Heinz type estimates for graphs in Euclidean space

Abstract

Abstract. Let M n be an entire graph in the Euclidean (n + 1)-space R n+1 . Denote by H, R and |A|, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of M n . We prove that if the mean curvature H of M n is bounded, then inf M |R| = 0, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of M n is negative, then inf M |A| = 0. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate inf |H|, inf |R| and inf |A| for graphs in R n+1 of C 2 real-valued functions defined on closed balls in R n .

Remarks on solitons for inverse mean curvature flow

“…Moreover, a detailed introduction to various Omori–Yau maximum principles and its applications can be found in [1]. We use the suitable distant function used in [9] to obtain that the Ricci curvature is bounded below, and we apply the Omori–Yau maximum principle to the translating soliton for IMCF. Theorem Let M be a connected and complete Riemannian manifold with Ricci curvature bounded below.…”

Finding umbilics on open convex surfaces

“…Theorem 1.7 in[5]). Let M n be the graph in R n+1 of a smooth function f defined on a closed ball of radius r in R n .…”

On the complexity of isometric immersions of hyperbolic spaces in any codimension

“…For the sake of completeness, we mention the well-known problem that H m cannot be C 2 isometrically immersed in R 2m−1 , although this conjecture is not the focus of the present work (indeed, our results are valid in arbitrary codimension). For background on this problem, as well as related works, see [4], [5], [11], [14], [16] - [20]).…”