Related papers: Frankel property and Maximum Principle at Infinity…

It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this result can be found in the literature, for…

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface $M$ into $\mathbb R^n$, $n\geq 3$. It follows that the inclusion of the space of such immersions into the space of all nonflat…

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two ``splitting in a finite cover'' theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the…

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

We say that a Riemannian manifold M has rank at least k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns-Spatzier, later generalized by Eberlein-Heber, states that a complete,…

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon…

This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension four, all prime compact parametrized minimal surfaces within $M$ have self-intersections in general position in the following sense:…

We study/construct (proper and non-proper) Morse functions on complete Riemannian manifolds, the level hypersurfaces of which have positive mean curvatures at all non-critical points. We show, for instance, that if a complete Rieannin…

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover,…

We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

We construct sequences of `expander manifolds' and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander…

This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by C. LeBrun in…

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies…

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher…

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly…

In this paper, we develop the infinitesimal geometry of the limit spaces of compact Riemannian manifolds with boundary, where we assume lower bounds on the sectional curvatures of manifolds and boundaries and the second fundamental forms of…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…