arXiv research

Strong Frankel theorem for shrinkers in all dimensions.

Proves Frankel theorem for generic submanifolds in Sasakian manifolds.

In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.

New curvature condition helps characterize Kähler manifolds.

In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle in \cite{BS2} proposed by Brendle and Schoen.

The paper proves a stronger Frankel theorem for minimal hypersurfaces of $S^k$.

A classical theorem of Frankel for compact Kähler manifolds states that a Kähler S^1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then Frankel's theorem still holds. Finally, we present several concrete situations in w…

In this paper we obtain theorems of Barth-Lefschetz type in Sasakian geometry. As corollaries, this implis connectedness principle and Frankel's type theorem.

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDu…

We prove some structure results for \emph{transverse reducible} Sasaki manifolds. In particular, we show Sasaki manifolds with positive Ricci curvature is transversely irreducible, and so there is no join (product) construction for irregular Sasaki-Einstein manifolds, as opposed to the quasi-regular case done by Wang-Z…

Rigidity theorem for critical points of Allen-Cahn equation on S³.

We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of ${k}$-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^{2}$-topology.

The paper proves a theorem about mean curvature in Euclidean and hyperbolic spaces.

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we …

The study generalizes curvature bounds for manifolds with boundary.

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…

The aim of this paper is to give a proof the Frankel conjecture by using the Kahler Ricci flow alone without assuming apriori the existence of Kahler Einstein metrics. However, there is an essential difference between the real case and the Kahler case. I didn't realize this difference in the calculation of the previous…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of a…

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…

The paper explores properties of 1-surfaces in hyperbolic space and proves strong half-space theorems.

The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.

Study shows universal circle isomorphic to flow space ideal boundary.

In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the Kähler-Ricci flow.

Study minimal hypersurfaces in manifolds with bounded Ricci curvature.

Study rigidity of PSU(1,1) actions on circle via harmonic measures.

The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof …

New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.

Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bund…

Two minimal hypersurfaces in a ball intersect in any half-ball.

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniquenes…

In this paper, we study any Kähler manifold where the positive orthogonal bisectional curvature is preserved on the Kähler Ricci flow. Naturally, we always assume that the first Chern class $C_1$ is positive. In particular, we prove that any irreducible Kähler manifold with such property must be biholomorphic to $\math…

An alternate proof shows how foliation extensions work in 3D spaces.

Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.

The paper classifies adjacencies in $L^\infty$-Delaunay triangulations of abelian differentials.

We first show that for a bounded pseudoconvex domain with a manifold quotient of finite-volume in the sense of Kahler-Einstein measure, the identity component of the automorphism group of this domain is semi-simple without compact factors. This partially answers an open question in [Fra95]. Then we apply this result in…

In hyperbolic L-spaces, we find multiple pseudo-Anosov flows with unique properties.

The paper constructs infinitely many surfaces with specific mean curvature.

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B…

Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \}$. This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked …

The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.

Unified approach to various energy conditions in spacetime geometry.

Positive curvature manifolds from smaller ones using division algebras and geodesic flow.

AI generates theorems and proofs for training theorem provers.

Global inverse function theorem proved easily using Riemannian geometry.