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Register a new userBlow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation
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We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of lower energy solutions to the above equation fails when the dimension of the manifold is not less than $62$.
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We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $ngeq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+infty$.
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(ngeq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$ with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Lis results, we answer this conjecture affirmatively except for the case that $ngeq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.
We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the link of the singular set. Within this class of conic metrics, we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.
A version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol$_3$ is proved. Moreover, we construct examples of admissible domains where the results may be applied.
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