The second generalized Yamabe invariant and conformal mean curvature flow on manifolds with boundary
Introduction
The following Uniformization Theorem is classical:
Theorem 1.1 Suppose is a 2-dimensional compact Riemannian manifold without boundary. Then there exists a Riemannian metric conformal to g such that the Gaussian curvature of is constant.
One would like to study the possible generalization of the Uniformization Theorem in higher dimensions. One has the following in [29]:
Yamabe problem without boundary: Given an n-dimensional compact Riemannian manifold without boundary such that , find a metric conformal to g such that its scalar curvature is constant.
The conformal Laplacian is defined as where is the Laplacian of g and is the scalar curvature of g. Solving the Yamabe problem without boundary is equivalent to finding a positive smooth solution u of for some constant C. To obtain a solution of (1.2), Yamabe [29] defined the following quantity, called Yamabe invariant: where In particular, if u is a positive smooth function such that , then u is a solution of (1.2) and the scalar curvature of is constant. The following theorem was proved by Aubin [6] and Trudinger [28]:
Theorem 1.2 (i) There holds where is the n-dimensional unit sphere and is the standard Riemannian metric in . (ii) Furthermore, if the inequality in (1.3) is strict, i.e. , then the Yamabe problem without boundary is solvable for .
The Yamabe problem without boundary was solved by Trudinger [28], Aubin [6], and Schoen [25]. Moreover, one can consider the generalization of Uniformization Theorem on manifolds with boundary.
Uniformization Theorem with boundary: Given a 2-dimensional compact Riemannian manifold with smooth boundary ∂M,
(Ia) there exists a metric conformal to g such that its Gaussian curvature is constant in M and its geodesic curvature vanishes on ∂M;
(Ib) there exists a metric conformal to g such that its Gaussian curvature vanishes in M and its geodesic curvature is constant on ∂M.
The existence of these metrics was proved by Osgood-Phillips-Sarnak [24]. In view of this, one would also like to consider the generalization of the above Uniformization Theorem with boundary in higher dimensions. Again there are two types:
Yamabe problem with boundary: Given an n-dimensional compact Riemannian manifold with smooth boundary ∂M such that ,
(IIa) find a metric conformal to g such that its scalar curvature is constant in M and its mean curvature vanishes on ∂M;
(IIb) find a metric conformal to g such that its scalar curvature vanishes in M and its mean curvature is constant on ∂M.
These have been studied by many people. See [1], [2], [11], [14], [15], [16], [22], [23] and the references therein. In this paper, we focus on the Yamabe problem with boundary (IIb). In addition to the conformal Laplacian, one can define the following operator in an n-dimensional compact Riemannian manifold with smooth boundary ∂M: where is the outward normal derivative with respect to g, and is the mean curvature of g. We call the boundary operator of g. Solving the Yamabe problem with boundary (IIb) is equivalent to finding a positive smooth solution u of for some constant C. It is clear that a critical point of the functional is a solution of (1.5). We remark that the functional defined here is different from the one defined in [15] by a constant multiple. Sometimes we also write as J, when the background metric g is clear from the context. Similar to the Yamabe problem without boundary, one can define the generalized Yamabe invariant:
Let be the n-dimensional unit disk, i.e. Then equipped with the standard flat metric is an n-dimensional Riemannian manifold with boundary The following theorem was proved by Escobar, which is the corresponding version of Theorem 1.2.
Theorem 1.3 Proposition 2.1 in [15] (i) There holds (ii) On the other hand, if the inequality in (1.7) is strict, i.e. , the Yamabe problem with boundary (IIb) is solvable for .
For an n-dimensional compact Riemannian manifold without boundary, the conformal Laplacian is a self-adjoint operator, hence has eigenvalue , i.e. for some . When , we have (see [5]) where is the conformal class of g. This motivates Ammann and Humbert to define and study the k-th Yamabe invariant in [5]: We remark that the k-th CR Yamabe invariant, a CR analogue of the k-th Yamabe invariant, has been defined and studied in [7] and [20].
Inspired by the work of Ammann and Humbert in [5], we define and study the k-th generalized Yamabe invariant for manifolds with boundary. See sections 2, 3 and 7. We then, in sections 4 and 5, estimate the second generalized Yamabe invariant for the manifolds with boundary. In section 6, we prove that, under some conditions, the second generalized Yamabe invariant can be attained by a generalized metric (see section 2 for the definition of the generalized metric).
As another approach to tackle the Yamabe problem without boundary, Hamilton [18] introduced the Yamabe flow on manifolds without boundary: Here, is the average of the scalar curvature of : See [9], [10], [13], [26], [31] for results related to the Yamabe flow on manifolds without boundary.
In [8], Brendle introduced Yamabe-type flows on manifolds with boundary. More precisely, to tackle the Yamabe problem with boundary (IIa), Brendle introduced the following Yamabe flow with boundary: This has been studied in [4], [8], [12]. On the other hand, for the Yamabe problem with boundary (IIb), Brendle introduced the following type of flow, which we call conformal mean curvature flow: Here, is the average of the mean curvature of : This has been studied in [3], [8], [12].
The following conformal Schwarz lemma was first proved by Yau [30]:
Theorem 1.4 Suppose is a compact Riemannian manifold without boundary whose scalar curvature satisfies where and are constants, and is the Yamabe metric conformally equivalent to g with scalar curvature . Then we have Moreover, we have the following: (i) if and only if ; (ii) if and only if .
Note that for two conformal metrics and , we say that if and . In [27], Suárez-Serrato and Tapie used a Yamabe-type flow to reprove Theorem 1.4. See also [19] for a CR version of Theorem 1.4.
Inspired by Theorem 1.4, we prove in section 8 the following conformal Schwarz lemma for manifolds with boundary by using a flow similar to (1.11):
Theorem 1.5 Let be an n-dimensional compact Riemannian manifold with smooth boundary ∂M satisfying where and are constants. Let be the unique metric in the conformal class of g with in M and on ∂M. Then we have Moreover, we have the following: (i) if and only if on ∂M; (ii) if and only if on ∂M.
Denote the volume of ∂M with respect to g by , i.e. . By integrating (1.12), we get the following volume rigidity.
Theorem 1.6 With the same assumptions in Theorem 1.5, there holds Moreover, we have the following: (i) if and only if ; (ii) if and only if .
Section snippets
Variational characterization of and the generalized metrics
Let be an n-dimensional compact Riemannian manifold with smooth boundary ∂M, where . The boundary operator defined in (1.4) has eigenvalues i.e. there exists for each k such that Inspired by the definition of the k-th Yamabe invariant in (1.8), we define the k-th generalized Yamabe invariant: It follows from the definition that the k-th generalized Yamabe invariant depends only
The Euler-Lagrange equation
In this section, we will prove the following Euler-Lagrange equation of a minimizer of .
Theorem 3.1 Euler-Lagrange equation Assume that and that is attained by a generalized metric with and u is positive almost everywhere on ∂M. Let be as in Proposition 3.3. Then, . In particular, Moreover, w has alternating sign and for all .
To prove Theorem 3.1, we need the following: Lemma 3.2 Let and . We assume that
Lower bound for
In this section, we give a lower bound of by relating to .
Theorem 4.1 There holds . Proof For all and such that , we have the functional which is continuous on . We define for all . To prove Theorem 4.1, it follows from Proposition 2.2 that it suffices to show that for all
Upper bound for
We have the following relation between and . Here be the disjoint union of two copies of the n-dimensional unit disk with their standard flat metric .
Proposition 5.1 . Proof Let be an arbitrary metric on . We write for the first copy in and for the second copy in . There holds It
Existence of a minimizer of
In this section, we prove that is attained by a generalized metric under certain conditions.
Theorem 6.1 Let be a compact Riemannian manifold with smooth boundary ∂M such that . Then, is attained by a generalized metric in the following cases: Proof To prove Theorem 6.1, we consider a sequence of metrics , where , which minimizes
The k-th Yamabe invariant
Proposition 7.1 Suppose that . Then . Proof We can assume that . There exist smooth functions satisfying such that Let be fixed and choose a system of Fermi coordinates at p. Let η be a cutoff function such that , , and , where is defined as before. Define the function where and
Conformal mean curvature flow
Let be an n-dimensional compact Riemannian manifold with smooth boundary ∂M. Suppose that the scalar curvature of g and the mean curvature of g satisfy where and are constants.
We say that a family of metrics is increasing conformal mean curvature flow, denoted by , if it is a solution of the following evolution equation: where . Note that this is a
Appendix
First, we give the proof of Lemma 3.2. We let be a fixed number and be a large number which will tend to ∞. We also let . Then we define the following functions for : and It is easy to check the following properties of and : (see (A.1)-(A.3) in [5]) For all , we have Now we are ready to prove Lemma 3.2.
Proof of Lemma 3.2
Acknowledgements
The authors would like to thank the anonymous referee for his very careful reading of the manuscript and his valuable comments which helped to improve the manuscript. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019041021), and by Korea Institute for Advanced Study (KIAS) grant funded by the Korea government (MSIP).
References (31)
Convergence of scalar-flat metrics on manifolds with boundary under a Yamabe-type flow
J. Differ. Equ.
(2015)- et al.
The second Yamabe invariant
J. Funct. Anal.
(2006) Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary
J. Funct. Anal.
(2003)Rigidity in a conformal class of contact form on CR manifold
C. R. Math. Acad. Sci. Paris
(2015)- et al.
Extremals of determinants of Laplacians
J. Funct. Anal.
(1988) A compactness theorem for scalar-flat metrics on manifolds with boundary
Calc. Var. Partial Differ. Equ.
(2011)Blow-up phenomena for scalar-flat metrics on manifolds with boundary
J. Differ. Equ.
(2011)- et al.
Convergence of the Yamabe flow on manifolds with minimal boundary
Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5)
(2020) Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
J. Math. Pures Appl. (9)
(1976)- et al.
The second CR Yamabe invariant
Math. Nachr.
(2017)
A generalization of the Yamabe flow for manifolds with boundary
Asian J. Math.
Convergence of the Yamabe flow for arbitrary initial energy
J. Differ. Geom.
Convergence of the Yamabe flow in dimension 6 and higher
Invent. Math.
An existence theorem for the Yamabe problem on manifolds with boundary
J. Eur. Math. Soc.
Conformal curvature flows on a compact manifold of negative Yamabe constant
Indiana Univ. Math. J.
Cited by (2)
THE WEIGHTED YAMABE FLOW WITH BOUNDARY
2023, Communications on Pure and Applied AnalysisResults related to the transverse Yamabe problem
2022, International Journal of Mathematics