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Conformal Metrics of Constant Scalar Curvature and Constant Boundary Mean Curvature Xuezhang Chen Department of Mathematics�� Nanjing University�� Nanjing�� China Abstract This is a survey of recent results of the author and his collaborators on the existence and compactness/non-compactness of conformal metrics with constant scalar curvature and constant boundary mean curvature. We mainly focus our efforts on proving a conjecture proposed by Zheng-Chao Han and Yanyan Li in [16]�� which is a refined version of the boundary Yamabe problem. 1 Motivation The boundary Yamabe problem was initially studied by Escobar in 1992�� scalar flat with constant boundary mean curvature problem in [11] and constant scalar curvature with minimal boundary problem in [12]�� respectively. Readers may refer to [7] and references therein for more comments on these two problems. The problem of the existence of conformal metrics with (nonzero) constant scalar curvature and (nonzero) constant boundary mean curvature was also raised by Escobar [10] in 1996�� and a closely related Sobolev trace inequality in the halfspace was also established in Escobar [13]. A refined version was proposed by Zheng-Chao Han and Yanyan Li [16] in the following way: Conjecture 1(Han-Li�� 1999) Let (M�� g0) be an n-dimensional smooth compact Riemannian manifold of positive Yamabe constant with boundary and*�� thenthere exists a conformal metric such that its scalar curvature equals and its boundary mean curvature equals any real number c. Furthermore�� Han-Li confirmed their conjecture in several cases: and the boundary admits at least one non-umbilic point; see [15]. and (M�� g0) is locally conformally flat with umbilic boundary; see [16]. [14]. However�� this conjecture is far less developed after Han-Li��s work. Around 2015�� we began to investigate the boundary Yamabe problem and related topics�� including Han-Li��s conjecture. Many new technical tools (e.g.�� conformal Fermi coordinates introduced by Marques [17] etc.) and methods (especially�� the curvature flow approach) have emerged after the resolution of the Yamabe problem. This enables us to make some further developments on this conjecture�� which also brings some natural applications in conformal geometry and in establishing new geometric inequalities. For instance�� some new geometric inequalities on Poincaré-Einstein manifolds are discovered and rigidity theorems in the quality cases are established in Chen-Lai-Wang [6]. 2 Preliminary Let (M�� g0) be a smooth compact Riemannian manifold of dimension with boundary . be the conformal Laplacian and hg0 be the first order boundary operator�� where ��0 = ��g0 is the outward unit normal on M�� Rg0 and hg0 be the scalar curvature and the boundary mean curvature with respect to g0�� respectively. The pair (Lg0 ��Bg0) is conformally covariant: For any and with. (2.1) We introduce two types of the (generalized) Yamabe constants by respectively. It is not hard to know that Y if and only if. Readers refer to [11�� 12�� 8�� 7] for more properties of the above Yamabe constants. The Han-Li��s conjecture is equivalent to the solvability of positive solutions to PDE: For all (2.3) We only focus on a compact manifold of positive Yamabe constant with boundary�� which is the most interesting case. A boundary bubble is defined by. Here en is the unit direction vector in n-th coordinate and. Then W. satisfies. (2.4) Before presenting our main results�� we need to set up some notations. For . We define where Wg0 is the Weyl tensor in M and ��g(sh��)0 is the second fundamental form on .M with its the trace-free part . 3 Constrained variational problems We define the scalar curvature and mean curvature functional by . Consider the minimizing problem under either of the following constraints: i. Homogeneous constraint: ii. Non-homogeneous constraint: It is not hard to verify that any positive smooth minimizer of the above problem can provide a conformal metric with constant scalar curvature and constant boundary mean curvature. 3.1 Existence of minimizers: Subcritical approximations Since the method of subcritical approximations has been successfully used to solve the Yamabe problem�� it is natural to be applied to this critical growth problem. For the homogeneous constraint�� the strategy of subcritical approximations method is as follows. and Clearly�� it follows from (2.1) that Ya�� is also a conformal invariant. Step 1. A criterion of the existence of the minimizers of Ya��b. for any . A direct method in calculus variation derives the existence of positive smooth minimizers for Qqa��b[u] in the subcritical case �� with the help of the strict geometric inequality below we can overcome the difficulty in the critical case�� due to the loss of compactness of the embedding of Sobolev and trace inequalities. Proposition 2 If for any a�� b �� R+�� then can be achieved