arXiv research

Proves a new convergence theorem for mean curvature flow in spheres.

The paper proves sphere theorems for submanifolds in Kähler manifolds.

Sphere theorems for submanifolds in Kähler and Sasaki spaces.

Computer proof verifies key differential properties in sphere classifications.

Sharp convergence theorem for sphere submanifolds proved.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

Survey on mean curvature flow with sphere theorems and Yau rigidity theory.

In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton'…

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.

The 61-stem in stable homotopy groups of spheres is trivial.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci…

Paper finds explicit expressions for Jenkins-Strebel differentials on a sphere with four poles.

This is a survey paper focusing on the interplay between the curvature and topology of a Riemannian manifold. The first part of the paper provides a background discussion, aimed at non-experts, of Hopf's pinching problem and the Sphere Theorem. In the second part, we sketch the proof of the Differentiable Sphere Theore…

Sphere theorems proved for manifolds with specific curvature conditions.

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constan…

Minimal normal curvature immersions in the unit ball studied.

Paper proves existence of a CMC hypertorus in 4D sphere using numerical methods.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomo…

The paper classifies submanifolds in a sphere with a special tensor.

Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

The study extends Obata's theorem and classifies Finsler manifolds with transnormal functions.

This paper formalizes the h-principle and sphere eversion in differential topology.

The paper proves manifold diffeomorphism under certain curvature conditions.

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinchi…

The paper studies mean curvature flow of submanifolds in complex projective spaces.

Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.

A differential operator introduced by A. Gray on the unit sphere bundle of a Kähler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a Kähler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms among…

New proof shows minimal submanifolds of sphere are totally geodesic.

The study restricts manifolds with certain explicit SGL maps and constructs them.

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound $n(n-1)$ is critical in the sense that the hypersurface may contain geode…

The study identifies unique fluid flow patterns.

Research on surfaces in Laguerre geometry, focusing on L-isothermic, L-minimal, and generalized L-minimal surfaces.

Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.

Proves a theorem connecting graph theory spheres, reformulating Morse conditions.

Study of SU(2)-structures on tangent sphere bundles, discovering new metrics and structures.

The paper establishes a new sphere theorem for certain types of manifolds.

The paper develops a stability theorem for certain differential equations and applies it to free boundary problems.

We study almost complex surfaces in the nearly Kähler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic differential on every almost complex surface, and show that when this differentia…

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…

Sphere theorems for specific manifolds with curvature constraints.

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow the submanifold contracts smoothly to a round point in finite time. As a consequen…

Proves a Hopf theorem for non-constant mean curvature spheres.

Sphere theorem extended without Ricci curvature positivity.

Simplified proof of Lefschetz theorem for PL spheres.

In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for subma…