Related papers: Lu's conjecture for minimal surfaces

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros…

We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface…

We prove that in the three dimensional sphere with a bumpy metric or a metric with positive Ricci curvature, there exist at least four distinct embedded minimal two-spheres. This confirms a conjecture of S. T. Yau in 1982 for bumpy metrics…

The fundamental gap conjecture was recently proven by Andrews and Clutterbuck: for any convex domain in $\R^n$ normalized to have unit diameter, the difference between the first two Dirichlet eigenvalues of the Laplacian is bounded below by…

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form,…

In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory…

Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the $\pi_2$-systole of $M$ (i.e. the minimal achievable area of homotopically…

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. In this note, we put forward a conjecture that we prove is actually equivalent: every oriented simple graph…

Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in…

In this paper, we prove a pinching theorem for $n-$dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: $ |…

Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture…

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $\sigma_2(\kappa(X))=f(X, \nu(X)).$ In this note, we give a simpler proof of this result, and…

In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in CP^n relative to a possibly degenerate set of hyperplanes. In this paper, we generalize the Nochka's theorem to the case of curves in a complex projective…

Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of…

We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of…

This paper is the second in a two-part solution to Almgren's conjecture on the existence of area-minimizing submanifolds with fractal singular sets. In part one, we construct area-minimizing submanifolds with fractal singular sets on…

For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, i.e., a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our…

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2\pi$.…

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of…