In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space formMm(c), c 6= 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

Many optimization problems formulated on Riemannian manifolds involve their intrinsic Riemannian squared distance function. A notorious and important example is the centroid computation of a given finite constellation of points. In such setups, the implementation of the fast intrinsic Newton optimization scheme requires the availability of the Hessian of this intrinsic function. Here, a method ...

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [17, 21] for a priori estimates of the corresponding Monge-Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. W...

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.

Second-order methods for neural network optimization have several advantages over methods based on first-order gradient descent, including better scaling to large mini-batch sizes and fewer updates needed for convergence. But they are rarely applied to deep learning in practice because of high computational cost and the need for model-dependent algorithmic variations. We introduce a variant of ...

In this paper, we study stochastic non-convex optimization with non-convex random functions. Recent studies on non-convex optimization revolve around establishing second-order convergence, i.e., converging to a nearly second-order optimal stationary points. However, existing results on stochastic non-convex optimization are limited, especially with a high probability second-order convergence. W...

Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.

Abstract We prove quantitative versions for several results from geometric partial differential equations. Firstly, we obtain a double stability theorem Serrin’s overdetermined problem in spaceforms. Secondly, theorems Brendle’s Heintze–Karcher inequality respectively constant mean curvature classification class of warped product spaces. The key tool is the first author’s recent development lev...

We prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions the eigenvalues of complex Hessian matrix defining function and Levi curvatures hypersurface. Then we follow Reilly approach to an isoperimetric inequality. As application, obtain ``Soap Bubble Theorem'' star-shaped domains with positive constant bounding classical mea...

— We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the defo...