# zbMATH — the first resource for mathematics

Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. (English) Zbl 1175.53056
In the paper under review the authors prove two very interesting theorems motivated by a conjecture of Min-Oo: Let $$(M^n, g)$$ $$(n\geq 2)$$ be a compact Riemannian manifold with nonempty boundary $$\Sigma =\partial M$$. Suppose 4mm
{$$\bullet$$}
$$\text{Ric}\geq (n-1)g$$,
{$$\bullet$$}
$$(\Sigma, g|_\Sigma)$$ is isometric to the standard sphere $$\mathbb S^{n-1}\subset \mathbb R^n$$,
{$$\bullet$$}
$$\Sigma$$ is convex in $$M$$ in the sense that its second fundamental form is nonnegative. Then $$(M^n, g)$$ is isometric to the hemisphere $$\mathbb S^{n-1}=\{x\in\mathbb R^{n+1}: |x|=1, x_{n+1}\geq 0\}\subset \mathbb R^{n+1}$$.
and
Let $$(M^n, g)$$ be a compact Riemannian manifold with nonempty boundary $$\Sigma =\partial M$$ and $$\overline\Omega\subset \mathbb S^n_+$$ is a compact domain with smooth boundary in the open hemisphere. Suppose 4mm
{$$\bullet$$}
$$\text{Ric}\geq (n-1)g$$,
{$$\bullet$$}
there is an isometric embedding $$\iota: (\Sigma, g_\Sigma) \to \partial \overline\Omega$$
{$$\bullet$$}
$$\Pi \geq \Pi_0\circ \iota$$, where $$\Pi$$ is the second fundamental form of $$\Sigma$$ in $$M$$ and $$\Pi_0$$ is the second fundamental form of $$\partial \overline\Omega$$ in $$\mathbb S^n_+$$.

##### MSC:
 53C24 Rigidity results 58J32 Boundary value problems on manifolds
##### Keywords:
hemisphere; rigidity; Ricci curvature; conformal change
Full Text:
##### References:
  Eichmair, M.: The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere. arXiv math.DG/07063483v2 · Zbl 1187.53038  Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition · Zbl 1042.35002  Hang, F., Wang, X.: Rigidity and non-rigidity results on the sphere. Commun. Anal. Geom. 14, 91–106 (2006) · Zbl 1119.53029  Klingenberg, W.: Riemannian Geometry, 2nd edn. de Gruyter, Berlin (1995) · Zbl 0911.53022  Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2002)  Petersen, P.: Riemannian Geometry, 2nd edn. GTM, vol. 171. Springer, Berlin (2006) · Zbl 1220.53002  Reilly, R.: Applications of the Hessian operator in a Riemannian manifold. Indiana Math. J. 26(3), 459–472 (1977) · Zbl 0391.53019  Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979) · Zbl 0405.53045  Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981) · Zbl 0494.53028  Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Somerville (1994) · Zbl 0830.53001  Shi, Y., Tam, L.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002) · Zbl 1071.53018  Toponogov, V.: Evaluation of the length of a closed geodesic on a convex surface. Dokl. Akad. Nauk SSSR 124, 282–284 (1959) (Russian) · Zbl 0092.14603  Witten, E.: A simple proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981) · Zbl 1051.83532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.