The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
-
Jeffrey S. Case
,
Yi Wang
Abstract
In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset
1 Introduction
We view
Let
whose boundary is given by:
the Clifford torus.
On
where
where
Initially, we assume that our boundary data
where
Theorem 1.1
If
and
Some of the arguments follow those in [16], where (1.2) is solved on flag domains of
A general result of Nhieu [14] implies the existence of a solution
Theorem 1.2
If
for all
The key observation in the proof of Theorem 1.2 is that the single-layer potential extends to a bounded linear map
Our work is motivated in part by the desire to formulate and study the CR boundary Yamabe problem (cf. [3]): Given a closed CR three-manifold with boundary
One way to construct such a contact form is to show that there is a smooth, positive function that realizes the CR boundary Yamabe constant:
If
The positivity of
of
In Section 7, the equivalence of the signs of
Proposition 1.3
Let
Proposition 1.3 will be proved by observing that a minimizer
where
Since the standard contact form on
Theorem 1.4
Let
where
is the standard Gaussian hypergeometric function. Then
The solution of Theorem 1.4 is normalized so that
This article is organized as follows. In Section 2, we give some background in pseudohermitian geometry and Fourier analysis, in the specific setting of the Clifford torus. In Section 3, we compute the single-layer potential in local coordinates. Theorems 1.1 and 1.2, Proposition 1.3, and Theorem 1.4 will be proved in Sections 4, 5, 6, and 7, respectively.
2 Setup and notations
2.1 Pseudohermitian geometry
A CR three-manifold is a pair
A (strictly pseudoconvex) pseudohermitian manifold is a triple
An admissible coframe for
Note that
Let
where
We say that
Given a function
where
It is readily computed (cf. [12, equation (2.4)]) that
Suppose that
Suppose now that
where
Since the Tanaka-Webster connection preserves the contact form and the Levi form,
The CR Yamabe operator
The CR Yamabe operator is conformally covariant [10, equation (3.1)]: If
When the contact form is clear from context, we shall write
The conformal transformation law [11, Lemma 3.4] for the Tanaka-Webster connection implies that if
It follows that the CR Yamabe functional
is CR invariant; indeed,
for all positive
2.2 The standard sphere
S
3
Let
be the contact form on
Then
so
we have
It follows that
so the Tanaka-Webster scalar curvature is
where
The fundamental solution to
2.3 The Clifford torus
Recall the domain
Its boundary is the Clifford torus
and
The Reeb vector field on
For
and
so that
and the above expressions for
Furthermore,
so
2.4 Fourier analysis
To parametrize
and the associated abelian group
and it gives rise to a corresponding norm function on
The Lebesgue measure
An orthonormal basis of
and hence Parseval’s formula reads
where
The convolution of two functions
for every
Thus
2.5 Parametrizing
Σ
t
For
One can see that the above map is a well-defined bijection by noting that
is a bijection (Figure 2). We have
![Figure 1
The dotted square is
[
−
π
,
π
]
2
{\left[-\pi ,\pi ]}^{2}
. The tilted rectangle is the image of
(
u
,
v
)
∈
[
−
π
,
π
]
×
[
−
π
∕
2
,
π
∕
2
]
↦
(
v
+
u
,
v
−
u
)
∈
R
2
\left(u,v)\in \left[-\pi ,\pi ]\times \left[-\pi /2,\pi /2]\mapsto \left(v+u,v-u)\in {{\mathbb{R}}}^{2}
. By translating the colored pieces in the tilted rectangle either vertically or horizontally by
2
π
2\pi
, we see that the tilted rectangle also parametrizes a copy of
(
R
∕
2
π
Z
)
2
{\left({\mathbb{R}}/2\pi {\mathbb{Z}})}^{2}
.](/document/doi/10.1515/ans-2022-0072/asset/graphic/j_ans-2022-0072_fig_001.jpg)
The dotted square is
The dots represent the lattice
Under this parametrization of
In particular, setting
3 The single-layer potential
For
Given
satisfies
Proposition 3.1
For
exists at every
where
To prove this, first we understand the kernel
so from
where for
We collect a few facts we will need of
Lemma 3.2
If
If
In fact,
for all
For all
and hence
where the implicit constant is independent of
We have
(3.3)
We postpone the proof of this lemma to the end of this section.
Proof of Proposition 3.1
Using (2.4) and (3.2), if we write
whenever
Similarly, when
is the convolution of
for every
Now let
Furthermore, from (3.4) and our bounds for
Hence, to prove Proposition 3.1, by L’Hôpital’s rule and (3.5), it suffices to show, under the assumption that
exists and equals
To do so, we rewrite the expression inside the limit in (3.6) as
By (3.3) in Lemma 3.2, as
Proof of Lemma 3.2
First, for fixed
To proceed further, we prove that
In fact, the former is equivalent to
which implies, since
As a result,
If
This proves (a). If
This proves (b).
To prove (c), note that
near
then by a Taylor expansion
where
uniformly in
(d) follows from
and the lower bound
Finally, we prove (3.3) in (e). Since
From
where
and
We do so by Taylor expanding the numerators and denominators of the integrands. By choosing
Similarly, for
and
Thus
The error term involving
where we performed a change of variables
Similarly,
Finally, use
We Taylor expand the numerator by noting
and
where we abbreviated
and
We also Taylor expand the denominator, yielding
Thus
The second limit is zero by the same argument of the proof of (3.12). In the first limit, we are considering the limit as
We have
and using a change of variable
Thus
as desired.□
4 The classical solution to the Neumann boundary value problem
In this section, we prove Theorem 1.1.
4.1 Existence
We are now going to invert
Proposition 4.1
We have
Furthermore, if
The main thrust is in showing that
In the following, we prove that
Indeed, since
where
Thus we need to show that
Since
To bound this integral and establish (4.1), it will be convenient to approximate
where
We approximate
respectively, so that the homogeneous function
will be a good approximation of
for all
whereas
and
To carry out the aforementioned strategy, we differentiate
where we have, via the half-angle formulae, the following expressions for the numerators:
Similarly, we write
where
Later, we will use
In fact, we have
and
Thus, Taylor expansion (up to orders 5 and 3, respectively) gives
and
As a result, we can estimate the main terms for the expressions for
and
Furthermore, the error terms can be bounded using
and
The above eight inequalities together imply (4.4), (4.5), and (4.6).
Now let
For
and
By homogeneity, we then have
Furthermore,
where
with
Indeed, when
where we first used (4.3), then used
Applying this with
we obtain
Inequality (4.8) now follows since
and since we can use (4.10) with
We also note that
because then we can apply (4.9) and use
Using similar techniques, we can prove, for
and
where
and
This is because when
where in the first inequality we applied (4.4), and in second inequality we used (4.10) with
where
The numerical values of
In the next two lemmas, we will prove two bounds for
Lemma 4.2
Suppose that
where
and
Proof
Define
We estimate the first term by putting absolute value inside, and we integrate by parts in the second term using
We integrate by parts once more for the last double integral, using
Taking absolute values, and using
The first four integrals are all contained in
All the integrals on the right-hand side of the above equation are now integrals of homogeneous functions of
Similarly, the integrals involving
Finally, by the Fundamental Theorem of Calculus and the chain rule,
which, by homogeneity, is equal to
Thus
and putting all these together, one sees that
Remembering
Lemma 4.3
Suppose that
where
and
Proof
The proof is similar to the previous lemma (and slightly easier); one only needs to split the integral into two parts, one where
More precisely, define
We estimate the first term by putting absolute value inside, and we integrate by parts in the second term using
Thus
The first three integrals are all contained in
All the integrals on the right-hand side of the above equation are now integrals of homogeneous functions of
which, by homogeneity, is equal to
Thus
and by scaling all other integrals similarly one sees that
Remembering
We apply the above two lemmas with
Corollary 4.4
Suppose m and n are positive integers with the same parity. If
Proof
We apply Lemma 4.2 with
so
Next, recall
Together with the numerical value of
Finally, we set
Corollary 4.5
Suppose m and n are positive integers with the same parity. If
Proof
We apply Lemma 4.3 with
so
Next, recall
Together with the numerical value of
Finally, we set
Remark 4.6
The Mathematica codes used in Corollaries 4.4 and 4.5 are enclosed in Appendix A for interested readers.
From Corollary 4.4, if
The right-hand side is an increasing function of
The points for which
From Corollary 4.5, if
The right-hand side is an increasing function of
It remains to verify that
5 The weak solution to the Neumann problem
In this section, we prove Theorem 1.2.
5.1 Functional analysis
Recall that the Folland-Stein space
is continuous. Therefore the trace operator
Since
is the continuous extension of
Necessarily,
i.e.,
for all
5.2 Potential theory
Let
We have
if
Proposition 5.1
If
Proof
Note that
by the bound for
Thus
But
If
so
The numerator of the first term is even in
Plugging
where
To summarize, if
where
It follows that
is essentially the sum of the convolutions of
where we applied Young’s convolution inequality on
uniformly in
uniformly in
To prove the claim, by Parseval on
uniformly in
for all
all uniformly in
To proceed further, define
Proposition 5.2
To prove this, write
where
(Recall
Proposition 5.3
For
for
Proof of Proposition 5.3
In fact, if
where
and
Using the bound
it is easy to see that
in fact,
and
Furthermore, using the bound
one can show that
This is because
Finally, we claim that for almost every
In fact, the operators on both sides of the inequality are bounded on
Recall
be the convolution of
In fact, if
On the other hand, if
with the last integral being both zero because
This proves (5.5), and hence
But
where
This proves (5.4).□
We are now ready to complete the proof of Theorem 1.2. We need to show that if
in the weak sense, i.e.,
for all
and
Our earlier estimate for
for almost every
The measure of the last set with respect to
Letting
Our earlier estimate for
This shows that
For uniqueness, we need to show that if
then
Setting
6 Proof of Proposition 1.3
Let
First, since
for all
where
Lemma 6.1
For any
Here we identify a function on
if
Proof of Lemma 6.1
To prove the lemma, note that if
where
for all
In fact, since
by writing
and integrating by parts in the second integral using either
Now, if
Recall now our critical point
Using the bound from Section 4, we have
Putting these together, we see that
for all
7 The boundary Yamabe contact form on
Σ
We conclude this article by proving our results about the boundary Yamabe constant.
First, we prove that the boundary Yamabe constant
Proposition 7.1
Let
Proof
Set
Jerison proved [9, Theorem 7.1] that
Suppose first that
Suppose next that
Suppose finally that
Combining Jerison’s regularity results [9, Theorem 7.1] with Bony’s maximum principle [1, Corollaire 3.1] implies that
In particular, there are positive constants
for all
Therefore
Second, we prove that
Proposition 7.2
Let
Proof
Denote
Suppose first that
Therefore
Suppose next that
The strong maximum principle [1, Corollaire 3.1] implies that
We conclude from the Sobolev trace embedding theorem that
Finally, we construct a scalar flat contact form on
Proof of Theorem 1.4
Introduce new coordinates
making the obvious identifications. A straightforward computation yields
It readily follows that
We conclude that a smooth function
By the change of variables
It is well known [15, §14.2 and §14.8] that the space of solutions of (7.4) is spanned by the Legendre function:
Rewriting this in terms of
-
Funding information: Case was partially supported by the Simons Foundation (Grant #524601). Chen was partially supported by NSF Award DMS-3103392. Wang was partially supported by NSF Career Award DMS-1845033. Yang was partially supported by the Simons Foundation (Grant #1006518). Yung was partially supported by a Future Fellowship FT200100399 from the Australian Research Council.
-
Conflict of interest: The authors state no conflict of interest.
Appendix A
The Mathematica codes used in Corollaries 4.4 and 4.5 are reproduced below for reference. In order to simplify expressions and outputs in Mathematica, some definitions in the codes are different from the definitions taken in the rest of this article. We first point out the relations between these two sets of definitions and then list the codes:
Appendix B
We use the following codes in the Mathematica to verify the inequality
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© 2023 the author(s), published by De Gruyter
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