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Register a new userTotal mean curvature of the boundary and nonnegative scalar curvature fill-ins
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In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question ref{extension1}). Then we introduce a fill-in invariant (see Definition ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromovs conjectures formulated in cite{Gro19} (see Conjecture ref{conj0} and Conjecture ref{conj1} below)
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In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Sigma,gamma,H)$. We prove that given a metric $gamma$ on $mathbf{S}^{n-1}$ ($3leq nleq 7$), $(mathbf{S}^{n-1},gamma,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $gamma$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $int_{mathbf S^{n-1}}H,mathrm dmu_gamma$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.,23 in cite{Gromov4}). In the second part of this paper, we investigate the $theta$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
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 $62$.
We study harmonic maps from a 3-manifold with boundary to $mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.
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.
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