Convergence rate of the weighted conformal mean curvature flow

Shota Hamanaka; Pak Tung Ho

doi:10.1515/agms-2025-0026

Artikel Open Access

Convergence rate of the weighted conformal mean curvature flow

Shota Hamanaka und Pak Tung Ho

Veröffentlicht/Copyright: 31. Juli 2025

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Aus der Zeitschrift Analysis and Geometry in Metric Spaces Band 13 Heft 1

Abstract

In this article, we study the convergence rate of the following Yamabe-type flow

R ϕ ( t ) m = 0 in M and ∂ ∂ t g ( t ) = 2 ( h ϕ ( t ) m − H ϕ ( t ) m ) g ( t ) ∂ ∂ t ϕ ( t ) = m ( H ϕ ( t ) m − h ϕ ( t ) m ) on ∂ M

on a smooth metric measure space with boundary ( M , g ( t ) , e − ϕ ( t ) d V g ( t ) , e − ϕ ( t ) d A g ( t ) , m ) , where R ϕ ( t ) m is the weighted scalar curvature, H ϕ ( t ) m is the weighted mean curvature, and h ϕ ( t ) m is the average of the weighted mean curvature.

Keywords: flows related mean curvature; PDEs on manifolds; smooth metric measure spaces

MSC 2010: Primary 53E10; Secondary 53C23; 35R01

1 Introduction

Suppose M is a compact, n -dimensional manifold without boundary, where n ⩾ 3 , and g 0 is a Riemannian metric on M . As a generalization of the Uniformization Theorem, the Yamabe problem is to find a metric conformal to g 0 such that its scalar curvature is constant. This was first introduced by Yamabe [32] and was solved by Trudinger [31], Aubin [3] and Schoen [29].

The Yamabe flow is defined as follows:

(1.1) ∂ ∂ t g ( t ) = − ( R g ( t ) − r g ( t ) ) g ( t ) ,

where R g ( t ) is the scalar curvature of g ( t ) and r g ( t ) is the average of R g ( t ) :

r g ( t ) = ∫ M R g ( t ) d V g ( t ) ∫ M d V g ( t ) .

This is a geometric flow first introduced by Hamilton in [17] to study the Yamabe problem. The convergence of the flow was proved in [5,6,14,30,34] under various assumptions.

In [8], Carlotto et al. studied the rate of convergence of the Yamabe flow (1.1). They proved the following:

Theorem 1.1

Assume g ( t ) is a solution of the Yamabe flow (1.1) that converges in C 2 , α ( M , g ∞ ) to g ∞ as t → ∞ for some α ∈ ( 0 , 1 ) . Then there is a δ > 0 depending only on g ∞ such that:

If g ∞ is an integrable critical point, then the convergence occurs at an exponential rate, that is
‖ g ( t ) − g ∞ ‖ C 2 , α ( M , g ∞ ) ≤ C e − δ t .
In general, the rate of convergence cannot be worse than polynomial, that is,
‖ g ( t ) − g ∞ ‖ C 2 , α ( M , g ∞ ) ≤ C ( 1 + t ) − δ .

Theorem 1.2

Assume that g ∞ is a nonintegrable critical point of the Yamabe energy with order of integrability p ≥ 3 . If g ∞ satisfies the Adams-Simon positive condition A S p , then there exists metric g ( 0 ) conformal to g ∞ such that the solution g ( t ) of the Yamabe flow (1.1) starting from g ( 0 ) exists for all time and converges in C ∞ ( M , g ∞ ) to g ∞ as t → ∞ . The convergence occurs “slowly” in the sense that

C − 1 ( 1 + t ) − 1 p − 1 ≤ ‖ g ( t ) − g ∞ ‖ C 2 , α ( M , g ∞ ) ≤ C ( 1 + t ) − 1 p − 2

for some constant C > 0 .

We refer the readers to [8, Definition 8] and [8, Definition 10], respectively, for the precise definitions of integrable critical point and Adams-Simon positive condition A S p . In [8], examples of g ∞ satisfying the A S p , for some p ≥ 3 , have been found. Hence, there exist examples of the Yamabe flow (1.1), which does not converge exponentially. This is the first geometric flow found not to have exponential convergence. (Before it is known that there are some flows which do not converge exponentially, but they are not geometric).

Now consider a compact, n -dimensional manifold M with smooth boundary ∂ M , where n ⩾ 3 , and g 0 is a Riemannian metric on M . One can still talk about the Yamabe problem for manifolds with boundary, and there are two types. For the first type, one would like to find a conformal metric g such that its scalar curvature R g is constant in M and its mean curvature H g vanishes on ∂ M . For the second type, one would like to find conformal metric g such that its scalar curvature R g vanishes in M and its mean curvature H g is constant on ∂ M . These problems have been studied by many authors. See [7,15,16] for example.

Similar to the Yamabe flow, Brendle introduced some geometric flows in [4] to study the Yamabe problem for manifolds with boundary. For the first type, the Yamabe flow with boundary is defined as follows:

(1.2) ∂ g ∂ t = − ( R g − r g ) g in M and H g = 0 on ∂ M .

Almaraz and Sun has considered in [2] the convergence of the flow (1.2). On the other hand, for the second type, the conformal mean curvature flow is defined as follows:

(1.3) ∂ g ∂ t = − ( H g − h g ) g on ∂ M and R g = 0 in M ,

where h g is the average of the mean curvature H g :

h g = ∫ ∂ M H g d A g ∫ ∂ M d A g .

In [1], Almaraz has studied the convergence of the flow (1.3). See also [13,18–21] for the results related to the flows (1.2) and (1.3).

In this article, in the same spirit of [33], we study the corresponding geometric flow (1.3) defined on smooth metric measure spaces with boundary. To state our results, we require some terminologies.

Definition 1.3

Let ( M , ∂ M , g ) be a Riemannian manifold with boundary ∂ M and let us denote by d V g and d A g the volume element induced by g on M and ∂ M , respectively. Set a function ϕ ∈ C ∞ ( M ) and a dimensional parameter m ∈ [ 0 , ∞ ) . In the case m = 0 , we require that ϕ = 0 . A smooth metric measure space with boundary is a five-tuple ( M , g , e − ϕ d V g , e − ϕ d A g , m ) .

The weighted scalar curvature R ϕ m of a smooth metric measure space with boundary ( M , g , e − ϕ d V g , e − ϕ d A g , m ) is

(1.4) R ϕ m ≔ R g + 2 Δ g ϕ − m + 1 m ∣ ∇ ϕ ∣ g 2 ,

where R g and Δ g are the scalar curvature and the Laplacian associated to the metric g , respectively. The weighted mean curvature is

(1.5) H ϕ m = H g + ∂ ϕ ∂ ν g ,

where H g and ∂ ∂ ν g are the mean curvature and the outward normal derivative with respect to g , respectively.

Conformal equivalence between smooth metric measure spaces are defined as follows. See [9] for more details.

Definition 1.4

Two smooth metric measure spaces with boundary ( M , g , e − ϕ d V g , e − ϕ d A g , m ) and ( M , g ˆ , e − ϕ ˆ d V g ˆ , e − ϕ ˆ d A g ˆ , m ) are pointwise conformally equivalent if there is a function 0 < w ∈ C ∞ ( M ) such that

(1.6) ( M , g ˆ , e − ϕ ˆ dvol g ˆ , e − ϕ ˆ d A g ˆ , m ) = ( M , w 4 m + n − 2 g , w 2 ( m + n ) m + n − 2 e − ϕ d V g , w 2 ( m + n − 1 ) m + n − 2 e − ϕ d A g , m ) ,

Note that in the case m = 0 , we have ϕ = 0 , and the aforementioned conformal equivalence is reduced to the classical definition of conformal equivalence.

Definition 1.5

Let ( M , g , e − ϕ d V g , m ) be a smooth metric measure space. The weighted Laplacian Δ ϕ : C ∞ ( M ) → C ∞ ( M ) is the operator defined as follows:

Δ ϕ ψ = Δ ψ − ⟨ ∇ ϕ , ∇ ψ ⟩ g for any ψ ∈ C ∞ ( M ) ,

It is formally self-adjoint with respect to the measure e − ϕ d V g . For more about smooth metric measure spaces, we refer the readers to [9–11].

Definition 1.6

Given a smooth metric measure spaces with boundary ( M , g , e − ϕ d V g , e − ϕ d A g , m ) , the weighted conformal Laplacian ( L ϕ m , B ϕ m ) is given by the interior operator and boundary operator

(1.7) L ϕ m = − Δ ϕ + n + m − 2 4 ( n + m − 1 ) R ϕ m in M , B ϕ m = ∂ ∂ ν g + n + m − 2 2 ( n + m − 1 ) H ϕ m on ∂ M ,

where ν g is the outward unit normal with respect to g .

Consequently, under the conformal equivalence of (1.6), the transformation law of the weighted scalar curvature and the weighted mean curvature [28, Proposition 1] are

(1.8) R ϕ ˆ m = 4 ( n + m − 1 ) n + m − 2 w − m + n + 2 m + n − 2 L ϕ m w in M , H ϕ ˆ m = 2 ( n + m − 1 ) n + m − 2 w − n + m n + m − 2 B ϕ m w on ∂ M .

Given a compact smooth metric measure space without boundary ( M , g , e − ϕ d V g , m ) , the weighted Yamabe problem is to find another smooth metric measure space ( M , g ˆ , e − ϕ ˆ d V g ˆ , m ) conformal to ( M , g , e − ϕ d V g , m ) such that its weighted scalar curvature R ϕ ˆ m is constant. Note that the weighted Yamabe problem in this article is different from that introduced by Case in [9]. See [10,12,27] for more results related to Case’s weighted Yamabe problem.

Similar to the Yamabe flow (1.1), the weighted Yamabe flow was introduced by Yan [33] to study the weighted Yamabe problem and is the evolution equation defined on ( M , g ( t ) , e − ϕ ( t ) d V g ( t ) , m ) :

∂ ∂ t g ( t ) = ( r ϕ ( t ) m − R ϕ ( t ) m ) g ( t ) , ∂ ∂ t ϕ ( t ) = m 2 ( R ϕ ( t ) m − r ϕ ( t ) m ) ,

where r ϕ ( t ) m is the average of the weighted scalar curvature R ϕ ( t ) m . In [33], Yan proved the convergence of the weighted Yamabe flow. The convergence rate of the weighted Yamabe flow was studied in [23].

One can consider the weighted Yamabe problem with boundary, and there are two types: Find ( M , g ˆ , e − ϕ ˆ d V g ˆ , e − ϕ ˆ d A g ˆ , m ) conformal to ( M , g , e − ϕ d V g , e − ϕ d A g , m ) such that

the weighted scalar curvature R ϕ ˆ m is constant in M and the weighted mean curvature H ϕ ˆ m is zero on ∂ M , or
the weighted scalar curvature R ϕ ˆ m is zero in M and the weighted mean curvature H ϕ ˆ m is constant on ∂ M .

See [24,28] and references therein for related results.

To study the weighted Yamabe problem with boundary (I), the second author and his collaborators have introduced in [24] the weighted Yamabe flow with boundary, which is the weighted version of (1.2):

(1.9) ∂ ∂ t g ( t ) = ( r ϕ ( t ) m − R ϕ ( t ) m ) g ( t ) ∂ ∂ t ϕ ( t ) = m 2 ( R ϕ ( t ) m − r ϕ ( t ) m ) in M and H ϕ ( t ) m = 0 on ∂ M ,

where r ϕ ( t ) m is the average of the weighted scalar curvature R ϕ ( t ) m . On the other hand, to study the weighted Yamabe problem with boundary (II), the weighted conformal mean curvature flow has been introduced in [25]:

(1.10) R ϕ ( t ) m = 0 in M and ∂ ∂ t g ( t ) = 2 ( h ϕ ( t ) m − H ϕ ( t ) m ) g ( t ) ∂ ∂ t ϕ ( t ) = m ( H ϕ ( t ) m − h ϕ ( t ) m ) ϕ ( t ) on ∂ M

where h ϕ ( t ) m is the average of the weighted mean curvature H ϕ ( t ) m .

In view of the results about convergence rate mentioned earlier, we study in this article the convergence rate of the weighted Yamabe flow with boundary (1.9) and the weighted conformal mean curvature flow (1.10). The following theorems are the first two main results of this article, which are about the weighted conformal mean curvature flow (1.10). We will deal with the weighted Yamabe flow with boundary (1.9) in Section 6.

Theorem 1.7

Assume that ( g ( t ) , ϕ ( t ) ) is a solution of the weighted conformal mean curvature flow (1.10) which converges in C 2 , α ( ∂ M , g ∞ ) to ( g ∞ , ϕ ∞ ) as t → ∞ for some α ∈ ( 0 , 1 ) . Furthermore, assume that ( g ∞ , ϕ ∞ ) has constant weighted mean curvature on ∂ M and vanishing weighted scalar curvature in M. Then there is a δ > 0 depending only on ( g ∞ , ϕ ∞ ) so that

If ( g ∞ , ϕ ∞ ) is an integrable critical point, then the convergence occurs at an exponential rate, that is,
‖ g ( t ) − g ∞ ‖ C 2 , α ( ∂ M , g ∞ ) ≤ C e − δ t
for some constant C > 0 depending on ( g ( 0 ) , ϕ ( 0 ) ) .
In general, the rate of convergence cannot be worse than exponential, that is,
‖ g ( t ) − g ∞ ‖ C 2 , α ( ∂ M , g ∞ ) ≤ C ( 1 + t ) − δ
for some constant C > 0 depending on ( g ( 0 ) , ϕ ( 0 ) ) .

From [4, Theorem 1.2] and [25], it is known that the limiting structure ( g ∞ , ϕ ∞ ) has constant weighted mean curvature and vanishing weighted scalar curvature if we assume one of the following:

m > 0 ,
the conformal class [ ( g ∞ , ϕ ∞ ) ] contains a smooth metric-measure structure with nonpositive weighted mean curvature and vanishing weighted scalar curvature, or
( m , ϕ ∞ ) = ( 0 , 0 ) , M is locally conformally flat, ∂ M is umbilic, and the boundary of the universal cover of M is connected.

Theorem 1.8

Assume that ( g ∞ , ϕ ∞ ) is a nonintegrable critical point of the energy E (which will be defined in Section 2) with order of integrability p ≥ 3 . If ( g ∞ , ϕ ∞ ) satisfies AS p , then there exists a smooth metric-measure structure ( g ( 0 ) , ϕ ( 0 ) ) conformal to ( g ∞ , ϕ ∞ ) such that the weighted conformal mean curvature flow ( g ( t ) , ϕ ( t ) ) starting at ( g ( 0 ) , ϕ ( 0 ) ) exists for all time and converges in C ∞ ( M , g ∞ ) to ( g ∞ , ϕ ∞ ) as t → ∞ . The convergence occurs “slowly” in the sense that

C − 1 ( 1 + t ) − 1 p − 2 ≤ ‖ g ( t ) − g ∞ ‖ C 2 , α ( M , g ∞ ) ≤ C ( 1 + t ) − 1 p − 2

for some constant C > 0 .

The following is the plan of this article: after recalling some definitions and preliminaries in Section 2, we prove Theorem 1.7 in Section 3. Theorem 1.8 will be proved in Section 4. In Section 5, we construct an example of smooth metric measure space, which satisfies the condition A S 3 . This allows us to conclude that there exists weighted conformal mean curvature flow that slowly converges exactly at a polynomial rate described in Theorem 1.8. In Section 6, we consider the convergence rate of the weighted Yamabe flow with boundary (1.9).

2 Definitions and preliminaries

Let ( M , g , e − ϕ d V g , e − ϕ d A g , m ) be a compact smooth metric measure space. The following sharp Sobolev trace inequality was proved by Li and Zhu (cf. [26, Theorem 0.1]): There exists a constant C = C ( M , g ) > 0 such that

(2.1) ∫ ∂ M ∣ f ∣ 2 ( n − 1 ) n − 2 d A g n − 2 n − 1 ≤ S ∫ M ∣ ∇ g f ∣ 2 d V g + C ∫ ∂ M f 2 d A g

for all f ∈ H 1 ( M , g ) , where S = 2 n − 2 ω n − 1 − 1 and ω n − 1 is the volume of the ( n − 1 ) -dimensional standard unit sphere S n − 1 . Since 2 ( n + m − 1 ) n + m − 2 ≤ 2 ( n − 1 ) n − 2 for m ≥ 0 , by Hölder’s inequality, we have

‖ f ‖ L n + m − 2 2 ( n + m − 1 ) ( ∂ M ) ≤ ‖ f ‖ L 2 ( n − 1 ) n − 2 ( ∂ M ) ⋅ ‖ 1 ‖ L p ( ∂ M )

where 1 ⁄ p = 2 ( n + m − 1 ) n + m − 2 − 1 − 2 ( n − 1 ) n − 2 − 1 . In particular, we have

∫ ∂ M ∣ f ∣ 2 ( n + m − 1 ) n + m − 2 d A g n + m − 2 2 ( n + m − 1 ) ≤ C ( M , g , m ) ∫ ∂ M ∣ f ∣ 2 ( n − 1 ) n − 2 d A g n − 2 2 ( n − 1 ) .

By (2.1) and the continuity of ϕ , we can conclude that there exist constants A and B depending only on ( M , g , e − ϕ d V g , e − ϕ d A g , m ) such that

(2.2) ∫ ∂ M ∣ f ∣ 2 ( n + m − 1 ) n + m − 2 e − ϕ d A g n + m − 2 2 ( n + m − 1 ) ≤ A ∫ M ∣ ∇ g f ∣ 2 e − ϕ d V g + B ∫ ∂ M f 2 e − ϕ d A g

for all f ∈ H 1 ( M , e − ϕ d V g ) .

Moreover, the weighted integration by parts is the following: for any smooth function f on M and smooth vector field X on M ,

(2.3) ∫ M ⟨ ∇ f , X ⟩ e − ϕ ∞ d V g ∞ = − ∫ M f div ϕ ∞ ( X ) e − ϕ ∞ d V g ∞ + ∫ ∂ M f ⟨ X , ν g ∞ ⟩ g ∞ e − ϕ ∞ d A g ∞ ,

where div ϕ ∞ ( X ) ≔ div g ∞ X − ⟨ ∇ ϕ ∞ , X ⟩ g ∞ .

Suppose ( M , g , e − ϕ d V g , e − ϕ d A g , m ) is conformal to ( M , g ∞ , e − ϕ ∞ d V g ∞ , e − ϕ ∞ d A g ∞ , m ) in the sense of (1.6). The energy functional E g ∞ , ϕ ∞ is defined as follows:

E g ∞ , ϕ ∞ ( g , ϕ ) ≔ ∫ M R ϕ m e − ϕ d V g + ∫ ∂ M H ϕ m e − ϕ d A g ∫ ∂ M e − ϕ d A g n + m − 2 n + m − 1 .

If w is the conformal factor between ( M , g , e − ϕ d V g , e − ϕ d A g , m ) and ( M , g ∞ , e − ϕ ∞ d V g ∞ , e − ϕ ∞ d A g ∞ , m ) , i.e.,

( M , g , e − ϕ d V g , e − ϕ d A g , m ) = ( M , w 4 n + m − 2 g ∞ , w 2 ( n + m ) n + m − 2 e − ϕ ∞ d V g ∞ , w 2 ( n + m − 1 ) n + m − 2 e − ϕ ∞ d A g ∞ , m ) ,

then the energy functional E g ∞ , ϕ ∞ can be written as follows:

E g ∞ , ϕ ∞ ( g , ϕ ) ≕ E g ∞ , ϕ ∞ ( w ) = 4 ( n + m − 1 ) n + m − 2 ⋅ ∫ M w L ϕ ∞ m ( w ) e − ϕ ∞ d V g ∞ + ∫ ∂ M w B ϕ ∞ m ( w ) e − ϕ ∞ d A g ∞ ∫ ∂ M w 2 ( n + m − 1 ) n + m − 2 e − ϕ ∞ d A g ∞ n + m − 2 n + m − 1 .

Here,

L ϕ ∞ m ( w ) = − Δ ϕ ∞ w + n + m − 2 4 ( n + m − 1 ) R ϕ ∞ m w in M , B ϕ ∞ m ( w ) = ∂ w ∂ ν g ∞ + n + m − 2 2 ( n + m − 1 ) H ϕ ∞ m w on ∂ M , R ϕ ∞ m = R g ∞ + 2 Δ g ∞ ϕ ∞ − m + 1 m ∣ ∇ ϕ ∣ g ∞ 2 , H ϕ ∞ m = H g ∞ + ∂ ϕ ∞ ∂ ν g ∞ .

Thus, by using (1.8) and (2.3), one can calculate the differential of E g ∞ , ϕ ∞ as follows:

(2.4) 1 2 D E g ∞ , ϕ ∞ ( w ) [ v ] = ∫ M R ϕ m w m + n + 2 m + n − 2 v e − ϕ ∞ d V g ∞ + ∫ ∂ M ( H ϕ m − e ϕ m ) w n + m n + m − 2 v e − ϕ ∞ d A g ∞ ∫ ∂ M w 2 ( n + m − 1 ) n + m − 2 e − ϕ ∞ d A g ∞ n + m − 2 n + m − 1 ,

where

e ϕ m = E g ∞ , ϕ ∞ ( w ) ∫ ∂ M w 2 ( n + M − 1 ) n + m − 2 e − ϕ ∞ d A g ∞ .

Therefore, if a metric-measure structure ( g , ϕ ) is a critical point of E g ∞ , ϕ ∞ , then R ϕ m = 0 in M and H ϕ m = const on ∂ M .

From now on, we assume that ( M , g ∞ , e − ϕ ∞ d V g ∞ , e − ϕ ∞ d A g ∞ , m ) satisfies

R ϕ ∞ m ≡ 0 , H ϕ ∞ m ≡ c , and Vol ( ∂ M , g ∞ , ϕ ∞ ) ≔ ∫ ∂ M e − ϕ ∞ d A g ∞ = 1

for some constant c . We can always assume this, changing the scale of the metric if necessary. We denote ℋ ϕ ∞ by the space of all weighted harmonic functions in M , i.e.,

ℋ ϕ ∞ ≔ { v ∈ C 2 , α ( M , g ∞ ) : Δ ϕ ∞ v = Δ g ∞ v − ⟨ ∇ ϕ ∞ , ∇ v ⟩ g ∞ = 0 in M } ,

which is a Banach space equipped with the norm ‖ ⋅ ‖ C 2 , α ( M , g ∞ ) . Consider the space of functions

Ω ϕ ∞ ≔ 0 < w ∈ C 2 , α ( M , g ∞ ) : ∫ ∂ M w 2 ( n + m − 1 ) m + n − 2 e − ϕ ∞ d A g ∞ = 1 ∩ ℋ ϕ ∞ ,

and the corresponding conformal class

〚 g ∞ , ϕ ∞ 〛 1 ≔ w 4 n + m − 2 g ∞ , ϕ ∞ − 2 m m + n − 2 ln w : w ∈ Ω ϕ ∞ .

We define the following notations: for k ∈ N , we define the k th differential of the energy functional E g ∞ , ϕ ∞ on 〚 g ∞ , ϕ ∞ 〛 1 at the point w in the direction v 1 , … , v k by

D k E g ∞ , ϕ ∞ ( w ) [ v 1 , … , v k ] .

As we will see below, the functional v ↦ D k E g ∞ , ϕ ∞ ( w ) [ v 1 , … , v k − 1 , v ] is in the image of L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) under the natural embedding into ℋ ϕ ∞ ′ . Therefore, we will also write

D k E g ∞ , ϕ ∞ ( w ) [ v 1 , … , v k − 1 ]

for this element of L 2 ( ∂ M , e ∞ − ϕ d A g ∞ ) . When k = 1 , we will drop the (second) brackets, and thus consider D E g ∞ , ϕ ∞ ( w ) ∈ L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) .

Since R ϕ m = 0 in M and Vol ( M , g , ϕ ) = 1 for all ( g , ϕ ) ∈ ℋ ϕ ∞ , from (2.4), we may write the differential of E g ∞ , ϕ ∞ restricted to 〚 g ∞ , ϕ ∞ 〛 1 as follows:

(2.5) 1 2 D E g ∞ , ϕ ∞ ( w ) [ v ] = ∫ ∂ M ( H ϕ m − h ϕ m ) w n + m n + m − 2 v e − ϕ ∞ d A g ∞ .

Here, h ϕ m is the average of H ϕ m :

h ϕ m = ∫ ∂ M H ϕ m e − ϕ d A g ∫ ∂ M e − ϕ d A g .

Therefore, regarded as an element of L 2 ( ∂ M , e − ϕ ∞ d V g ∞ ) , we have

(2.6) 1 2 D E g ∞ , ϕ ∞ = 2 ( n + m − 1 ) n + m − 2 ⋅ ∂ w ∂ ν g ∞ + H g ∞ m w − h ϕ m w n + m n + m − 2 ,

which corresponds to (2.6) in [21].

We denote by

WCℳC 1 = ( g , ϕ ) : w = ( e ϕ ∞ − ϕ ) n + m − 2 2 m ∈ Ω ϕ ∞ and H ϕ m is constant ,

which is the set of all smooth metric-measure structures ( g , ϕ ) conformal to ( g ∞ , ϕ ∞ ) with R ϕ m = 0 , H ϕ m = const , and Vol ( ∂ M , g , ϕ ) = ∫ ∂ M e − ϕ d A g = ∫ ∂ M w 2 m n + m − 2 e − ϕ ∞ d A g ∞ = 1 . We define the linearized operator of E g ∞ , ϕ ∞ at ( g ∞ , ϕ ∞ ) , ℒ ∞ , by means of the formula

− 2 n + m − 2 ∫ ∂ M w ℒ ∞ v d A g ∞ ≔ 1 2 D 2 E g ∞ , ϕ ∞ ( g ∞ , ϕ ∞ ) [ v , w ] = 1 2 d d t t = 0 D E g ∞ , ϕ ∞ ( 1 + t v ) [ w ]

for v ∈ ℋ ϕ ∞ ∩ v : ∫ ∂ M v e − ϕ ∞ d A g ∞ = 0 . A direct computation shows

− 2 n + m − 2 ℒ ∞ v = 2 ( n + m − 1 ) n + m − 2 ∂ v ∂ ν g ∞ + H g ∞ m v − n + m n + m − 2 H g ∞ m v ,

which implies

ℒ ∞ v = − ( n + m − 1 ) ∂ v ∂ ν g ∞ + H ϕ ∞ m v on ∂ M .

We define Λ 0 ≔ ker ℒ ∞ ∩ ℋ ϕ ∞ . Then Λ 0 is finite-dimensional, since it is the eigenspace for the weighted Steklov eigenvalue H ϕ ∞ m ⁄ ( n + m − 1 ) . We will write ker ℒ ∞ ⊥ for the L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) , and Λ 0 ⊥ ≔ ker ℒ ∞ ⊥ ∩ ℋ ϕ ∞ . Here, given a function f defined on ∂ M , we have identified f with the weighted harmonic function f ˆ (i.e., Δ ϕ ∞ f ˆ = 0 ) in M for which f ˆ ∣ ∂ M = f . Then, by using (2.2), we can verify that the energy E g ∞ , ϕ ∞ is an analytic in the sense of [35, Definition 8.8] in the same way as the proof of [21, Lemma 2.1].

Since ℒ ∞ is formally self-adjoint with respect to e − ϕ ∞ d A g ∞ , replacing ℋ with ℋ ϕ ∞ , L 2 ( ∂ M , g ∞ ) with L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) , respectively, one can see that all the arguments from the line 15 on page 8 in [21] to the end of the page 9 in [21] also hold for the weighted conformal mean curvature flow. In particular, as in [21, Proposition 2.2], one can define the analytic map F : Λ 0 ∩ { v : ‖ v ‖ L 2 ( ∂ M , g ∞ ) → R } , and when g ∞ is a nonintegrable critical point, we may expand it in a power series

(2.7) F ( v ) = F ( 0 ) + ∑ j ≥ p F j ( v ) ,

where F j is a degree j homogenous polynomial on Λ 0 and p is chosen so that F p is nonzero. We call such p as the order of the integrability of g ∞ . Also, as in [21, Definition 2.3], one can define the integrability of g ∞ ∈ WCℳC 1 .

Definition 2.1

For g ∞ ∈ WCℳC 1 , we say that g ∞ is integrable if for all v Λ 0 , there is a path w ( t ) ∈ C 2 ( ( − ε , ε ) × M , g ∞ ) such that w ( t ) 4 n + m − 2 g ∞ ∈ WCℳC 1 and w ( 0 ) = 1 , w ′ ( 0 ) = v .

If Λ 0 = { 0 } , i.e., ℒ ∞ is injective, it is standard to call g ∞ a nondegenerate critical point. If this holds, g ∞ is automatically integrable in the aforementioned sense. On the other hand, if Λ 0 is nonempty, then we call g ∞ degenerate. Moreover, as in [21, Definition 2.5], one can define the Adams-Simon positivity condition, AS p .

Definition 2.2

We say that g ∞ satisfies the Adams-Simon positivity condition ( AS p ) (here p is the order of integrability of g ∞ ), for g ∞ , if it is nonintegrable and F p ∣ S k attains a positive maximum for some v ˆ ∈ S k ⊂ Λ 0 . Here, S k is the unit sphere in Λ 0 , where the inner product on Λ 0 comes from the L 2 inner product on the tangent space T 1 〚 g ∞ , ϕ ∞ 〛 .

Replacing L 2 ( ∂ M , g ∞ ) with L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) , and noting that ℒ ∞ is formally self-adjoint with respect to e − ϕ ∞ d A g ∞ , one can show as in [21, Appendix A] that, in the expression (2.7), we have F 1 = F 2 = 0 and

D 3 E g ∞ , ϕ ∞ ( 1 ) [ v , u , z ] = 2 − n + m n + m − 2 n + m n + m − 2 − 1 H ϕ ∞ m ∫ ∂ M v u z e − ϕ ∞ d A g ∞ = − 4 ( n + m ) ( n + m − 2 ) 2 H ϕ ∞ m ∫ ∂ M v u z e − ϕ ∞ d A g ∞ .

Therefore,

(2.8) F 3 ( v ) = − 4 ( n + m ) ( n + m − 2 ) 2 H ϕ ∞ m ∫ ∂ M v 3 e − ϕ ∞ d A g ∞ .

3 Proof of Theorem 1.7

As in [23, Proof of Proposition 3.1], one can check that the assumption in [8, Proposition 12] is still satisfied for W ≔ L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) and B ≔ C 2 , α ( ∂ M , g ∞ ) . As a result, one can show the following Łojasiewicz-Simon-type inequality:

Proposition 3.1

Suppose g ∞ ∈ WCℳC 1 . There are θ ∈ ( 0 , 1 2 ] , ε > 0 and C > 0 depending only on n , m , and ( g ∞ , ϕ ∞ ) such that for u ∈ Ω ϕ ∞ with ‖ u − 1 ‖ C 2 , α ( ∂ M , g ∞ ) < ε , then

h u 4 n + m − 2 g ∞ m − h g ∞ m 1 − θ ≤ C ‖ D E g ∞ , ϕ ∞ ( u 4 n + m − 2 g ∞ ) ‖ L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) .

If g ∞ is integrable, i.e., Λ 0 = { 0 } , then θ = 1 2 . If g ∞ is nonintegrable, then θ ∈ ( 0 , 1 p ] , where p is the order of integrability of g ∞ .

Now we are ready to prove Theorem 1.7.

Proof of Theorem 1.7

Similar to the proof of [21, Theorem 1.6], we have

d d t ( h g ( t ) m − h g ∞ m ) = − ( n + m − 2 ) ∫ ∂ M ( h ϕ ( t ) m − H ϕ ( t ) m ) 2 e − ϕ ( t ) d A g ( t ) ≤ − c ∫ ∂ M ( h ϕ ( t ) m − H ϕ ( t ) m ) 2 u ( t ) 2 ( n + m ) n + m − 2 e − ϕ ∞ d A g ∞ = − c ‖ D E g ∞ , ϕ ∞ ( g ( t ) ) ‖ L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) 2 ≤ − c ∣ h ϕ ( t ) m − h g ∞ m ∣ 2 − 2 θ .

Here, the first equality follows from (2.13) in [25], and the first inequality follows from the fact that ‖ u ( t ) − 1 ‖ C 2 , α ( ∂ M , g ∞ ) ≤ ε for t ≥ t 0 . Furthermore, the second equality follows from (2.5) and the last inequality follows from the Łojasiewicz-Simon-type inequality in Proposition 3.1. Here, we remark that c > 0 is a constant depending only on n , m , and ( g ∞ , ϕ ∞ ) , which may change from line to line.

With this, we can follow the arguments in the proof of [21, Theorem 1.6] to finish the proof.□

4 Slowly converging weighted conformal mean curvature flow

4.1 Projecting the weighted conformal mean curvature flow with estimates

The following lemma could be proved as [23, Lemma 4.1]; we omit the proof.

Lemma 4.1

Assume that ( g ∞ , ϕ ∞ ) satisfies AS p , i.e., F p ∣ S k achieves a positive maximum for some point v ˆ in the unit sphere S k ⊂ Λ 0 . Then, for any fixed T ≥ 0 , the function

φ ( t ) ≔ φ ( t , T ) = ( T + t ) − 1 p − 2 4 ( n + m − 2 ) p ( p − 2 ) F p ( v ˆ ) 1 p − 2 v ˆ

solves 4 n + m − 2 φ ′ + D F p ( φ ) = 0 .

As in p. 12 of [21], we can define the parabolic Hölder spaces. Then one can prove the similar estimates as [21, Lemma 4.2] for D 3 E g ∞ , ϕ ∞ by using (2.8), i.e.,

D 3 E g ∞ , ϕ ∞ ( 1 ) [ v , u , z ] = − 4 ( n + m ) ( n + m − 2 ) 2 H ϕ ∞ m ∫ ∂ M v u z e − ϕ ∞ d A g ∞ .

Replacing E 0 ⊥ ( w ) in [21, Lemma 4.3] with

E 0 ⊥ ( w ) ≔ proj Λ 0 D E g ∞ , ϕ ∞ ( u ) u − 2 n + m − 2 − D E g ∞ , ϕ ∞ ( u ) ,

one can prove the same statement as [21, Lemma 4.3]. Following the proof of [21, Proposition 4.4], one can prove the following:

Proposition 4.2

There exists T 0 > 0 , ε 0 > 0 and c > 0 , all depending on ( g ∞ , ϕ ∞ ) and v ˆ , such that the following holds: Fix T > T 0 . Then, for φ ( t ) as in Lemma 4.1 and w ∈ C 2 , α ( M × [ 0 , ∞ ) ) ∩ ℋ ϕ ∞ , there are functions E ⊤ ( w ) and E ⊥ ( w ) such that u ≔ 1 + φ + w ⊤ + Φ ( φ + w ⊤ ) + w ⊥ is a solution to the weighted conformal mean curvature flow if and only if

(4.1) 4 n + m − 2 ( w ⊤ ) ′ + D 2 F p ( φ ) w ⊤ = E ⊤ ( w ) ,

(4.2) ( w ⊥ ) ′ − ℒ ∞ w ⊥ = E ⊥ ( w ) .

Here, as long as ‖ w ‖ C 2 , α ≤ ε 0 , the error terms E ⊤ and E ⊥ satisfy the same estimates as [21, Proposition 4.4].

4.2 Solving the kernel-projected flow with polynomial decay estimates

By replacing D in p. 17 of [21] with

D ≔ 4 ( n + m − 2 ) p ( p − 2 ) F p ( v ˆ ) D 2 F p ( v ˆ ) ,

one can rewrite the kernel-projected flow as follows:

(4.3) 4 n + m − 2 v i ′ + μ i T + t v i = E i ⊤ ≔ E ⊤ ⋅ e i , i = 1 , … , k ,

where μ 1 , … , μ k is the eigenvalues of D , e i is the corresponding orthonormal basis in which D is diagonalized and v i ≔ w ⊤ ⋅ e i . Then, for a fixed γ ∉ { n + m − 2 4 μ 1 , … , n + m − 2 4 μ k } , by replacing Π 0 in [21, (4.18)] with

Π 0 ≔ span v ∈ Λ 0 : D v = μ v and μ > 4 n + m − 2 γ ,

one can prove the similar statement as [21, Lemma 4.5] as follows.

Lemma 4.3

For any T > 0 such that ‖ E ⊤ ‖ C 1 + γ 0 , α < ∞ , there is a unique u with u ( t ) ∈ Λ 0 , t ∈ [ 0 , ∞ ) , satisfying ‖ u ‖ C γ 0 < ∞ , proj Π 0 ( u ( 0 ) ) = 0 , and such that v i ≔ u ⋅ e i solves the system (4.3). Furthermore, we have the bound

‖ u ‖ C 1 , γ 0 , α ≤ C ‖ E ⊤ ‖ C 1 + γ 0 , α .

Here, the constant C does not depend on T.

4.3 Solving the kernel-orthogonal projected flow

As in [21, Subsection 4.3], we define the weighted norms

‖ u ‖ L q 2 = sup t ∈ [ 0 , ∞ ) [ ( T + t ) q ‖ u ( t ) ‖ L 2 ( ∂ M ) ] ,

where the L 2 norm is the spatial norm of u ( t ) on ∂ M , taken with respect to e − ϕ ∞ d A g ∞ . Also, we define the weighted Hölder norms ‖ ⋅ ‖ C q 2 , α as that of p. 20 of [21]. Define Λ ↓ and Λ ↑ in the same way as the definitions in p. 20 of [21]. Then, one can interpret Λ ↑ as follows:

Λ ↑ = span { φ ∈ C ∞ ( ∂ M ) : Δ ϕ ∞ φ = 0 in M , ℒ ∞ φ + δ φ = 0 , δ < 0 } .

From the spectral theory, L 2 ( ∂ M , e − ϕ ∞ d A g ∞ ) = Λ ↑ ⊕ Λ 0 ⊕ Λ ↓ , and Λ ↑ and Λ 0 are finite-dimensional. Following Proof of [21, Lemma 4.6], one can prove the corresponding statement of [21, Lemma 4.6].

4.4 Construction of a slowly converging flow

Define the norms ‖ ⋅ ‖ γ * and consider the Banach space

X ≔ { w : ‖ w ‖ γ * < ∞ }

as in p. 23 of [21]. Then, as in [21, Proposition 4.7] (see also [23, Proposition 4.7]), one can prove the following:

Proposition 4.4

Assume that ( g ∞ , ϕ ∞ ) satisfies AS p . We may fix a point where F p ∣ S k − 1 achieves a positive maximum and denote it by v ˆ . Define

φ ( t ) = ( T + t ) − 1 p − 2 4 ( n + m − 2 ) p ( p − 2 ) F p ( v ˆ ) 1 p − 2 v ˆ

as in Lemma 4.1. Then, there exist C > 0 , T > 0 , 1 p − 2 < γ < 2 p − 2 and u ( t ) ∈ C ∞ ( M × ( 0 , ∞ ) ) such that Δ ϕ ∞ u ( t ) = 0 in M , u ( t ) > 0 , g ( t ) ≔ u ( t ) 4 n + m − 2 g ∞ is a solution to the weighted conformal mean curvature flow and

‖ w ⊤ ( t ) + Φ ( φ ( t ) + w ⊤ ) + w ⊥ ( t ) ‖ γ * = ‖ u ( t ) − φ ( t ) − 1 ‖ γ * ≤ C .

Now, from Proposition 4.4, one can prove Theorem 1.8 by following the exact argument in the proof of [21, Theorem 1.7].

5 Examples satisfying AS 3

Let ( M , g M ) be a closed m -dimensional Riemannian manifold with zero scalar curvature, i.e., R g M ≡ 0 . Let B 3 be the three-dimensional unit ball, i.e. B 3 = { ( x 1 , x 2 , x 3 ) ∈ R 3 : x 1 2 + x 2 2 + x 3 2 ≤ 1 } . Equipped with the flat metric g 0 , B 3 is a three-dimensional Riemannian manifold with boundary ∂ B 3 = { ( x 1 , x 2 , x 3 ) ∈ R 3 : x 1 2 + x 2 2 + x 3 2 = 1 } . As shown in [21, Section 5], the product manifold M × B 3 equipped with the product metric g ⊕ c − 1 g 0 is an ( m + 3 ) -dimensional manifold with smooth boundary ∂ ( M × B 3 ) = M × ∂ B 3 ; its scalar curvature and mean curvature are, respectively, given by R g ⊕ c − 1 g 0 = 0 and H g ⊕ c − 1 g 0 = 2 c . Moreover, the function 1 ⊗ ( 2 x 1 2 − x 2 2 − x 3 2 ) is a Steklov eigenfunction corresponding to the Steklov eigenvalue 2 c .

Now suppose that ( N n , h , e − ϕ d V h , m ) is a smooth metric measure space (without boundary) with zero weighted scalar curvature. If we choose c > 0 such that 2 c ⁄ ( n + m + 2 ) = 2 c , then the smooth metric measure space with boundary

( N × M × B 3 , ∂ ( N × M × B 3 ) , h ⊕ g ⊕ c − 1 g 0 , e − ϕ d V h ⊕ g ⊕ c − 1 g 0 , e − ϕ d A h ⊕ g ⊕ c − 1 g 0 , m )

has zero weighted scalar curvature and constant weighted mean curvature given by

R ϕ m = 0 and H ϕ m = 2 c .

Moreover, it is degenerate, for Λ 0 consists of Steklov eigenfunctions with Steklov eigenvalue 2 c ⁄ ( n + m + 2 ) = 2 c , and 1 ⊗ 1 ⊗ ( 2 x 1 2 − x 2 2 − x 3 2 ) lies in Λ 0 . On the other hand, by Fubini’s theorem,

∫ ∂ ( N × M × B 3 ) ( 1 ⊗ 1 ⊗ ( 2 x 1 2 − x 2 2 − x 3 2 ) ) 3 e − ϕ d A h ⊕ g ⊕ c − 1 g 0 = ∫ N × M × ∂ B 3 ( 1 ⊗ 1 ⊗ ( 2 x 1 2 − x 2 2 − x 3 2 ) ) 3 e − ϕ d A h ⊕ g ⊕ c − 1 g 0 = Vol ( M , g ) ∫ N e − ϕ d V h ∫ ∂ B 3 ( 2 x 1 2 − x 2 2 − x 3 2 ) 3 d A c − 1 g 0 .

By the calculation in [21, p. 27], the last term is nonzero. This shows that

( N × M × B 3 , ∂ ( N × M × B 3 ) , h ⊕ g ⊕ c − 1 g 0 , e − ϕ d V h ⊕ g ⊕ c − 1 g 0 , e − ϕ d A h ⊕ g ⊕ c − 1 g 0 , m )

satisfies AS 3 .

6 Convergence rate of the weighted Yamabe flow with boundary

In this section, we investigate the convergence rate of the weighted Yamabe flow with boundary (1.9). The proof is almost the same as the proof for case of the weighted Yamabe flow appeared in [23], except we have to take care of the boundary condition. As a result, we only sketch the proof.

Let ( M n , g , e − ϕ d V g , e − ϕ d A g , m ) be a smooth metric measure space with smooth boundary ∂ M . The normalized energy is defined as follows:

(6.1) E ( g , ϕ ) = ∫ M R ϕ m e − ϕ d V g + 2 ∫ ∂ M H ϕ m e − ϕ d A g ∫ M e − ϕ d V g n + m − 2 n + m .

If ( g = w 4 n + m − 2 g ∞ , ϕ = ϕ ∞ − 2 m n + m − 2 ln w ) for some positive w ∈ C 2 ( M ) and smooth metric measure space ( M n , g ∞ , e − ϕ ∞ d V g ∞ , e − ϕ ∞ d A g ∞ , m ) , then the normalized energy (6.1) can be written as follows:

(6.2) E ( g , ϕ ) = E g ∞ , ϕ ∞ ( w ) ≔ 4 ( n + m − 1 ) n + m − 2 ∫ M w L ϕ ∞ m ( w ) e − ϕ ∞ d V g ∞ + ∫ ∂ M w B ϕ ∞ m ( w ) e − ϕ ∞ d A g ∞ ∫ M w 2 ( n + m ) n + m − 2 e − ϕ ∞ d V g ∞ n + m − 2 n + m ,

thanks to (1.8). Here, L ϕ ∞ m and B ϕ ∞ m , respectively, denote the interior and boundary operator respectively, which are defined in (1.7). Thus, by using (2.3) and (1.8), we can calculate the differential of E g ∞ , ϕ ∞ as follows:

D E g ∞ , ϕ ∞ ( w ) [ v ] = 2 ∫ M ( R ϕ m − e ¯ ϕ m ) w n + m + 2 n + m − 2 v e − ϕ ∞ d V g ∞ + 4 ∫ ∂ M H ϕ m w n + m n + m − 2 v e − ϕ ∞ d A g ∞ Vol ( M , g , ϕ ) n + m − 2 n + m ,

where e ¯ ϕ m = E g ∞ , ϕ ∞ ( w ) ⋅ ( Vol ( M , g , ϕ ) ) − 1 and

( M , g , e − ϕ d V g , e − ϕ d A g , m ) = ( M , w 4 n + m − 2 g ∞ , w 2 ( n + m ) n + m − 2 e − ϕ ∞ d V g ∞ , w 2 ( n + m ) n + m − 2 e − ϕ ∞ d A g ∞ , m ) .

Therefore, from the fundamental lemma of the calculus of variations, one can see that a critical point ( g = w 4 n + m − 2 g ∞ , ϕ = ϕ ∞ − 2 m n + m − 2 ln w ) to the functional E g ∞ , ϕ ∞ must have constant weighted scalar curvature R g , ϕ m and vanishing weighted mean curvature H g , ϕ m , which solves the weighted Yamabe problem with boundary for the first type. From now on, we assume that ( g ∞ , ϕ ∞ ) satisfies that

R ϕ ∞ m ≡ c in M , H ϕ ∞ m ≡ 0 on ∂ M , and Vol ( M , g ∞ , ϕ ∞ ) ≔ ∫ M e − ϕ ∞ d V g ∞ = 1

for some constant c ∈ R . We can always assume this, changing the scale of the metric if necessary.

Let N be the subspace of C 2 , α ( M , g ∞ ) , which satisfies the Neumann boundary condition, i.e.,

N g ∞ = w ∈ C 2 , α ( M , g ∞ ) : ∂ w ∂ ν g ∞ = 0 on ∂ M ,

where ν g ∞ is the outward normal differential with respect to g ∞ . Consider the following space of functions:

Ω g ∞ , ϕ ∞ = 0 < w ∈ N g ∞ : ∫ M w 2 ( n + m ) n + m − 2 e − ϕ ∞ d V g ∞ = 1

and the conformal class associated to g ∞ :

[ [ [ g ∞ , ϕ ∞ ] ] ] 1 ≔ w 4 n + m − 2 g ∞ , ϕ ∞ − 2 m m + n − 2 ln w : w ∈ Ω g ∞ , ϕ ∞ .

By replacing C 2 , α ( M , g ∞ ) by the space N g ∞ , one can see that all the arguments in [23] for the weighted Yamabe flow also hold for the weighted Yamabe flow with boundary.

For example, since H ϕ m = 0 on ∂ M and Vol ( M , g , ϕ ) = 1 for all ( g , ϕ ) ∈ Ω g ∞ , ϕ ∞ , we may write the differential of E g ∞ , ϕ ∞ restricted to [ [ [ g ∞ , ϕ ∞ ] ] ] 1 as follows:

1 2 D E g ∞ , ϕ ∞ ( w ) [ v ] = ∫ M ( R ϕ m − r ϕ m ) w n + m + 2 n + m − 2 v e − ϕ ∞ d V g ∞ ,

where r ϕ m denotes the average weighted scalar curvature of the smooth metric measure space ( M , g , e − ϕ d V g , e − ϕ d A g , m ) , i.e.,

r ϕ m = ∫ M R ϕ m e − ϕ d V g ∫ M e − ϕ d V g .

We denote by

CWSCℬ 1 = { ( g , ϕ ) ∈ [ [ [ g ∞ , ϕ ∞ ] ] ] 1 : R ϕ m is constant } ,

which is the set of all metric-measure structures ( g , ϕ ) conformal to ( g ∞ , ϕ ∞ ) (in the sense of Definition 1.4) with constant weighted scalar curvature, zero weighted mean curvature, and Vol ( M , g , ϕ ) = 1 . The linearized weighted Yamabe operator at g ∞ is similarly defined as follows:

− 4 n + m − 2 ∫ M w ℒ ∞ v e − ϕ ∞ d V g ∞ ≔ 1 2 D 2 E g ∞ , ϕ ∞ [ v , w ] = 1 2 d d t ( D E ( 1 + t w ) [ v ] ) t = 0

for v ∈ N g ∞ , and the same computation in [23, Appendix] shows that

ℒ ∞ v = ( n + m − 1 ) Δ ϕ ∞ v + R ϕ ∞ m v in M .

If we define Λ 0 ≔ ker ℒ ∞ ( ⊂ N g ∞ ) , then it follows from a classical spectral theory that Λ 0 is finite dimensional and it is the eigenspace of the weighted Laplacian Δ ϕ ∞ for the eigenvalue R ϕ ∞ m n + m − 1 .

As in [23, Definition 2.4], one can define the integrability of g ∞ ∈ CWSCℬ 1 . Moreover, as in [23, Definition 2.6], one can define the Adams-Simon positivity condition ( AS p ) (here, p is the order of integrability of g ∞ ), for g ∞ .

With all these, one can follow the arguments in [23] to prove the following.

Theorem 6.1

Assume that ( g ( t ) , ϕ ( t ) ) is a solution to the weighted Yamabe flow with weighted minimal boundary (1.9) that is converging in C 2 , α ( M , g ∞ ) to ( g ∞ , ϕ ∞ ) as t → ∞ for some α ∈ ( 0 , 1 ) . Furthermore, we assume that ( g ∞ , ϕ ∞ ) has a constant weighted scalar curvature in M and vanishing weighted mean curvature on ∂ M . Then, there is δ > 0 depending only on ( g ∞ , ϕ ∞ ) such that

If ( g ∞ , ϕ ∞ ) is an integrable critical point, then the convergence occurs at an exponential rate
‖ ( g ( t ) , ϕ ( t ) ) − ( g ∞ , ϕ ∞ ) ‖ C 2 , α ( M , g ∞ ) ≤ C e − δ t
for some constant C > 0 depending on ( g ( 0 ) , ϕ ( 0 ) ) .
In general, the convergence cannot be worse than a polynomial rate
‖ ( g ( t ) , ϕ ( 0 ) ) − ( g ∞ , ϕ ϕ ) ‖ C 2 , α ( M , g ∞ ) ≤ C ( 1 + t ) − δ
for some constant C > 0 depending on ( g ( 0 ) , ϕ ( 0 ) ) , where
‖ ( g ( t ) , ϕ ( 0 ) ) − ( g ∞ , ϕ ϕ ) ‖ C 2 , α ( M , g ∞ ) = ‖ g ( t ) − g ∞ ‖ C 2 , α ( M , g ∞ ) + ‖ ϕ ( t ) − ϕ ∞ ‖ C 2 , α ( M , g ∞ ) .

From [4, Theorem 1.1] and [22, Theorem 1.8], the limiting structure ( g ∞ , ϕ ∞ ) has a constant weighted scalar curvature and vanishing mean curvature if we assume one of the following:

m > 0 ,
the conformal class [ ( g ∞ , ϕ ∞ ) ] contains a smooth metric-measure structure with nonpositive weighted scalar curvature and vanishing weighted mean curvature, or
( m , ϕ ∞ ) = ( 0 , 0 ) , M is locally conformally flat and ∂ M is umbilic.

Theorem 6.2

Assume that ( g ∞ , ϕ ∞ ) is a nonintegrable critical point of the functional E defined in (6.1) with order of integrability p ≥ 3 . If ( g ∞ , ϕ ∞ ) satisfies the Adams-Simon positivity condition AS p (here, p is the order of integrability of g ∞ ), then there exists a metric-measure structure ( g ( 0 ) , ϕ ( 0 ) ) conformal to ( g ∞ , ϕ ∞ ) such that the weighted Yamabe flow with weighted minimal boundary ( g ( t ) , ϕ ( t ) ) starting from ( g ( 0 ) , ϕ ( 0 ) ) exists for all time and converges in C ∞ ( M , g ∞ ) to ( g ∞ , ϕ ∞ ) as t → ∞ . The convergence occurs “slowly” in the sense that

C ( 1 + t ) − 1 p − 2 ≤ ‖ ( g ( t ) , ϕ ( t ) ) − ( g ∞ , ϕ ∞ ) ‖ C 2 ( M , g ∞ ) ≤ C ( 1 + t ) − 1 p − 2

for some constant C > 0 .

We can follow the argument in [23, Appendix] to show that

(6.3) F 3 ( v ) = − 8 ( n + m + 2 ) ( n + m − 2 ) 2 ∫ M v 3 e − ϕ ∞ d V g ∞ .

We denote the real 2 n 2 -dimensional complex projective space equipped with the Fubini-Study metric by ( P n 2 , g F S ) . (We normalize the Fubini-Study metric so the map S 2 n 2 + 1 → P n 2 from the unit sphere is a submersion.) Then we know that R F S = 4 n 2 ( n 2 + 1 ) and λ 1 ( g F S ) = 4 ( n 2 + 1 ) (see [8, Section 5.1]). Suppose that ( M n 1 , g M , e − ϕ d V g M , e − ϕ d A g M , m ) is a smooth metric measure space with minimal boundary H ϕ m = 0 and constant weighted scalar curvature R ϕ m = λ 1 ( g F S ) ( n 1 + n 2 + m − 1 ) .

We consider the product smooth metric measure space ( M n 1 × P n 2 , g = g M ⊕ g F S , e − ϕ d V g M ⊕ g F S , e − ϕ d A g M ⊕ g F S , m ) with boundary ∂ ( M × P n 2 ) = ∂ M × P n 2 . (Of course, we regard ( P n 2 , g F S ) as the trivial metric measure space ( P n 2 , g F S , e − ψ d V g F S , e − ψ d A g F S , l ) with ψ ≡ 0 .) Hence the weighted scalar curvature of this space is

R g , ϕ m = ( R g M + R g F S ) + 2 Δ g M ϕ − m + 1 m ∣ ∇ ϕ ∣ g M 2 = R g M , ϕ m + R g F S = λ 1 ( g F S ) ( n + m − 1 ) ,

where n = n 1 + 2 n 2 is the dimension of the product metric measure space. Therefore, the space Λ 0 consists of eigenfunctions of Δ g , ϕ with eigenvalue R g , ϕ m ⁄ ( n + m − 1 ) = λ 1 ( g F S ) . From this, the product space ( M n 1 × P n 2 , g = g M ⊕ g F S , e − ϕ d V g M ⊕ g F S , e − ϕ d A g M ⊕ g F S , m ) is degenerate. Let v be an eigenfunction corresponding to the eigenvalue λ 1 , i.e.,

− Δ g F S v = λ 1 v .

It follows from the Courant nodal domain theorem that v does not change sign. By replacing v with − v if necessary, we may assume that v > 0 . Since ϕ is a smooth function on M , the function 1 ⊗ v ∈ C ∞ ( M × P n 2 ) is an eigenfunction of Δ g , ϕ with eigenvalue λ 1 ( g F S ) , i.e.,

( n + m − 1 ) Δ g , ϕ ( 1 ⊗ v ) + R g , ϕ m ( 1 ⊗ v ) = ( n + m − 1 ) Δ g F S v + ( n + m − 1 ) λ 1 ( g F S ) v = 0 .

On the other hand, it follows from [8, Section 5.1] that

∫ P n 2 v 3 d V g F S ≠ 0 .

Hence, we have

F 3 ( 1 ⊗ v ) = − 8 ( n + m + 2 ) ( n + m − 2 ) 2 R g , ϕ m ∫ M × P n 2 v 3 e − ϕ d V g M ⊕ g F S = − 8 ( n + m + 2 ) ( n + m − 1 ) ( n + m − 2 ) 2 λ 1 ( g F S ) ∫ M n 1 e − ϕ d V g M ∫ P n 2 v 3 d V g F S = − 32 ( n 2 + 1 ) ( n + m + 2 ) ( n + m − 1 ) ( n + m − 2 ) 2 ∫ M n 1 e − ϕ d V g M ∫ P n 2 v 3 d V g F S ≠ 0 .

Therefore, the space ( M n 1 × P n 2 , g = g M ⊕ g F S , e − ϕ d V g M ⊕ g F S , e − ϕ d A g M ⊕ g F S , m ) satisfies the AS 3 condition.

This allows us, via Theorem 6.2, to conclude the existence of slowly converging weighted Yamabe flow with the weighted minimal boundary.

Corollary 6.3

There exists a weighted Yamabe flow with weighted minimal boundary in the conformal class of a metric in M × P n , which converges to the given metrics exactly at a polynomial rate, as in Theorem 6.2.

7 Concluding remarks

In Theorem 1.2 and Section 5, we construct a solution to the weighted conformal mean curvature flow ( N × M × B 3 , g ( t ) , ϕ ( t ) , m ) that converges to h ⊕ g ⊕ c − 1 g 0 on N × M × B 3 as t → ∞ with polynomial convergence rate, as described in Theorem 1.2. We remark that we do not know whether such solution ( g ( t ) , ϕ ( t ) , m ) of the weighted conformal mean curvature flow is nontrivial in the sense that ϕ ( t ) is not constant for some t > 0 . We raise the following:

Problem 7.1

Is the example constructed in Section 5 nontrivial? If not, can we construct such a solution of the weighted conformal mean curvature flow with nontrivial weight?

Note that if the smooth metric measure space ( N n , h , e − ϕ d V h , m ) has nontrivial weight, i.e., ϕ is not constant, then the solution constructed in Section 5 is nontrivial, i.e., ϕ ( t ) is not equal to constant for all t ≥ 0 . To see this, suppose that ϕ ( t ) is constant for some t ≥ 0 . Then the weighted conformal mean curvature flow reduces to the conformal mean curvature flow (1.3) with ϕ ( t ) ≡ 0 for all t ≥ 0 . In particular, the limit must be a smooth metric measure space with constant weight. But this is a contradiction, since the limit is

( N × M × B 3 , ∂ ( N × M × B 3 ) , h ⊕ g ⊕ c − 1 g 0 , e − ϕ d V h ⊕ g ⊕ c − 1 g 0 , e − ϕ d A h ⊕ g ⊕ c − 1 g 0 , m ) ,

which has nontrivial weight.

In view of this, in order to answer Problem 7.1, it suffices to find a smooth metric measure space ( N n , h , e − ϕ d V h , m ) (without boundary) with zero weighted scalar curvature, i.e., R ϕ m = 0 and its weight ϕ is not constant.

Similarly, we raise the following problem regarding Corollary 6.3.

Problem 7.2

Is the example constructed in Corollary 6.3 nontrivial? If not, can we construct such a nontrivial weighted Yamabe flow with weighted minimal boundary?

Note that if the smooth metric measure space with boundary ( M n 1 , g M , e − ϕ d V g M , e − ϕ d A g M , m ) has nontrivial weight, then the solution of the weighted Yamabe flow with boundary constructed at the end of Section 6 has nontrivial weight.

Acknowledgments

The authors are grateful for the reviewers’ valuable comments that improved the manuscript.

Funding information: SH was supported by JSPS KAKENHI Grant Number 24KJ0153. PTH was supported by the National Science and Technology Council (NSTC), Taiwan, with grant Number: 112-2115-M-032-006-MY2.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. SH and PTH have written the manuscript.
Conflict of interest: The authors state no conflict of interest.
Consent for publication: The authors declare the consent for publication.
Data availability statement: Not applicable.

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Received: 2025-02-22

Accepted: 2025-06-04

Published Online: 2025-07-31

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flows related mean curvature; PDEs on manifolds; smooth metric measure spaces

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