A new monotone quantity in mean curvature flow implying sharp homotopic criteria

1 Introduction

We start with the following theorem and its corollary proved in [17].

Explicit examples of non-trivial homotopy classes in π n ⁢ ( S m ) when n ≥ m inspire interest to find local geometric conditions that determine the homotopy class of a map from S n to S m . The “stretch” or “1-dilatation” of such a map has been considered by Lawson [8], Hsu [7], Gromov [3], and DeTurck–Gluck–Storm [2]. In [4], Gromov proposed to study how the 𝑘-dilation of a map S n → S m is related to its homotopy class. In particular, he showed in [4, page 179 and corollary on page 230] (see also [5, Section 2]) that there exists a constant ε ⁢ ( n , m ) > 0 such that a map f : ( S n , g std ) → ( S m , g std ) with 2-dilation (see below) less than ε ⁢ ( n , m ) everywhere must be homotopically trivial. It was conjectured that ε ⁢ ( n , m ) can be chosen to be one, as a map with 1-dilation less than one is clearly homotopically trivial. Unbeknown to the authors of [17], the conjecture was affirmed by Corollary 1.2: a map f : ( S n , g std ) → ( S m , g std ) with 2-dilation less than one is exactly an area-decreasing map. By contrast, such a result does not hold for the 3-dilation by the work of Guth [6]; see stronger results about 𝑘-dilations therein.

Area-decreasing maps in the mean curvature flow arise in a rather different context. It was shown in [20] that the Gauss maps of a mean curvature flow form a harmonic map heat flow, and area-decreasing maps correspond to a convex region in the Grassmannian. It was proved in [17] that the area-decreasing condition is preserved by the mean curvature flow under curvature assumptions of the domain and the target manifolds (see [18, 21] for the case when the target space is 2-dimensional). There were several later extensions [9, 16, 1] of the results in [17] to allow more general curvature conditions, all based on the study of the evolution equation of the symmetric 2-tensor S [ 2 ] (see [17, Section 5], or Section 3 of this article), whose positivity corresponds to the area-decreasing condition. In this article, we consider the evolution equation (3.6) of the logarithmic determinant of S [ 2 ] . This novel formulation significantly improves our understanding of the ambient curvature terms involved in the evolution equation, which leads to the sharp homotopic criteria mentioned in the title. In particular, unlike previous works whose assumptions rely on the sectional curvatures of the domain and target manifolds, Theorem 1.4 relies on the Ricci curvatures. This enables us to prove Corollaries 1.5 and 1.6 regarding the complex projective spaces.

It is best to describe the area-decreasing condition and the new monotone quantity in terms of singular values. Recall, for a C 1 map f : ( Σ 1 , g 1 ) → ( Σ 2 , g 2 ) between Riemannian manifolds, the singular values of d ⁢ f at a point p ∈ Σ 1 are the non-negative square roots of eigenvalues of ( d ⁢ f | p ) T ⁢ ( d ⁢ f | p ) . Suppose Σ 1 is of dimension 𝑛 and the singular values are λ i , i = 1 , … , n ; then the 2-dilation of 𝑓 is defined to be max i < j ⁡ λ i ⁢ λ j . A map is said to be area-decreasing if the 2-dilation is less than one everywhere, or λ i ⁢ λ j < 1 for any i < j .

This article considers the mean curvature flow of the graph of 𝑓, Γ f , as a submanifold of Σ 1 × Σ 2 . Under suitable curvature assumptions on Σ 1 and Σ 2 and the area-decreasing condition, the flow remains the graph of a family of maps f t . The new monotone quantity is

where λ i are the singular values of f t .

We state the main theorems in the following.

The condition κ 1 ≥ 1 means that κ 1 ⁢ ( σ ) ≥ 1 for any σ ∈ Λ 2 ⁢ ( T ⁢ Σ 1 ) .

The first curvature condition means that Ric 1 ⁡ ( u , u ) ≥ Ric 2 ⁡ ( v , v ) for any unit vectors u ∈ T ⁢ Σ 1 and v ∈ T ⁢ Σ 2 . The second curvature condition means that κ 1 ⁢ ( σ 1 ) + κ 2 ⁢ ( σ 2 ) > 0 for any σ 1 ∈ Λ 2 ⁢ ( T ⁢ Σ 1 ) and σ 2 ∈ Λ 2 ⁢ ( T ⁢ Σ 2 ) .

We state some corollaries of homotopic criteria which follow from the above theorems. Consider the complex projective spaces endowed with the Fubini–Study metrics g FS . Our theorems imply the following results.

It is known that the homotopy class of a map in [ C ⁢ P n , C ⁢ P m ] is determined by the induced map on the cohomology groups [13] for n < m , and by the degree for n = m (see [11]). Since the identity map is not homotopically trivial, the area-decreasing condition in Corollary 1.5 is sharp.

Another corollary concerns π 2 ⁢ n + 1 ⁢ ( C ⁢ P n ) , which is isomorphic to π 2 ⁢ n + 1 ⁢ ( S 2 ⁢ n + 1 ) ≅ Z by the long exact sequence induced by the Hopf fibration S 1 → S 2 ⁢ n + 1 → C ⁢ P n .

The Hopf fibration S 2 ⁢ n + 1 → C ⁢ P n represents the generator of π 2 ⁢ n + 1 ⁢ ( C ⁢ P n ) ≅ Z . It is a straightforward computation to show that the Hopf fibration ( S 2 ⁢ n + 1 , g std ) → ( C ⁢ P n , g FS ) has singular values 0 (of multiplicity 1) and 1 (of multiplicity 2 ⁢ n ). With this understood, Corollary 1.6 is sharp as n → ∞ .

The paper is organized as follows. In Section 2, we review some properties of graphical mean curvature flow. In Section 3, the evolution equation of the logarithmic determinant of S [ 2 ] is derived. Section 4 is devoted to prove the main theorems, Theorems 1.3 and 1.4. In Section 5, we explain the implications of the main theorems.

2 Preliminaries

For two Riemannian manifolds ( Σ 1 , g 1 ) and ( Σ 2 , g 2 ) , consider the symmetric 2-tensor 𝑆 on Σ 1 × Σ 2 defined by

for any X , Y ∈ T ⁢ ( Σ 1 × Σ 2 ) . It is not hard to see that 𝑆 is parallel with respect to the Levi-Civita connection of the product metric. Let n = dim Σ 1 and m = dim Σ 2 .

For a map f : Σ 1 → Σ 2 , let Γ f be the graph of 𝑓. As in [17, Section 4], one can apply the singular value decomposition at any p ∈ Σ 1 to d ⁢ f | p to find orthonormal bases { e i } i = 1 n for T ( p , f ⁢ ( p ) ) ⁢ Γ f and { e α } α = n + 1 n + m for ( T ( p , f ⁢ ( p ) ) ⁢ Γ f ) ⟂ such that

where { λ i } i = 1 n are the singular values of d ⁢ f | p . Indeed, there exist orthonormal bases { u i } i = 1 n for T p ⁢ Σ 1 and { v j } j = 1 m for T f ⁢ ( p ) ⁢ Σ 2 such that

It is convenient to set λ i to be 0 when i > min ⁡ { n , m } .

Suppose that { Γ f t } is a mean curvature flow. Let F t be the embedding

Consider the tensor F t ∗ ⁢ S on Σ 1 , and denote its components by S i ⁢ j . From [17, (3.7)], F t ∗ ⁢ S obeys the following equation along the mean curvature flow:

which is in terms of an evolving orthonormal frame. Here, the Laplacian Δ is the rough Laplacian for 2-tensors of Γ f ⁢ ( t ) , and R k ⁢ i ⁢ k ⁢ α = ⟨ R ⁢ ( e k , e α ) ⁢ e i , e k ⟩ is the coefficient of the Riemann curvature tensor of Σ 1 × Σ 2 .

The (spatial) gradient of F t ∗ ⁢ S is computed on [17, p. 1121] as follows:

where h α ⁢ k ⁢ i = ⟨ ∇ ̄ e k ⁢ e i , e α ⟩ is the coefficient of the second fundamental form of Γ f ⁢ ( t ) in Σ 1 × Σ 2 .

In terms of the frame { e i } i = 1 n (2.1), F t ∗ ⁢ S is diagonal, and

It follows that the 2-positivity^[1] of F t ∗ ⁢ S is equivalent to f ⁢ ( t ) being area-decreasing.

3 The evolution equation of log ⁡ ( det S [ 2 ] )

In [17, Section 5], the authors introduce a tensor S [ 2 ] from F t ∗ ⁢ S , which can be viewed as a symmetric endomorphism on Λ 2 ⁢ ( T ∗ ⁢ Γ f ) . With respect to an orthonormal frame,

for any i < j and k < ℓ .

Suppose that, at a space-time point 𝑝, 𝑆 is diagonal, S i ⁢ j | p = S i ⁢ i ⁢ δ i ⁢ j . As in (2.1), at this point,

Note that S i ⁢ i 2 + C i ⁢ i 2 = 1 . The fact that S [ 2 ] is positive is equivalent to S i ⁢ i + S j ⁢ j > 0 for i < j . We need the following calculation lemma for the evolution of S i ⁢ i + S j ⁢ j , i < j .

We are now ready to derive the evolution equation of log ⁡ ( det S [ 2 ] ) .