arXiv research

Paper proves pinching theorem for minimal surfaces in spheres.

In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous $3$-manifolds $E(κ,τ)$, $τ\neq0$. As an application, we give a rigidity result in the case of $κ> 4τ^2$ for the cmc surfaces under a pinching assumption of the second fundamental form.

We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold $\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with $f=\frac {t^2}{4}$. Also we classify closed $f$-minimal h…

Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.

In this paper, we study a slant submanifold of a complex space form. We also obtain an integral formula of Simons' type for a Kaehlerian slant submanifold in a complex space form and apply it to prove our main result.

Equations of Simons type are presented. They are satisfied by a pair of special operators associated to the immersion $Σ^2\looparrowright M^2(c)\times\mathbb{R}$ with constant mean curvature. Some immersions are characterized.

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

We prove an Atiyah-Bott-Berline-Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern-Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the uni…

Formula found for surfaces in Sol_3, leading to gap results.

Formula for spacelike submanifolds in warped products.

Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions. In this way, we show phenomenons of duality between Subman…

We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.

The paper studies special Lagrangian submanifolds in complex spaces and derives inequalities and flow methods.

Study on minimizing singular capillary cones with stability and instability results.

Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

We refine a metric bunching estimate for pinched manifolds.

Paper proves Hamilton's pinching theorem using mean curvature flow.

Flat Yang-Mills connections on pinched manifolds.

The paper pinches curvature in expanding Ricci solitons.

This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.

Compact shrinkers with curvature pinching conditions proven.

Alternative proof of flatness for Ricci-pinched 3-manifolds.

Proves CLT for Brownian paths on pinched negative curvature manifolds.

The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.

In this article, we generalize the classical Bochner-Weitzenböck theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degre…

Study pinched submanifolds in space forms, proving rigidity results.

Harmonic maps between pinched Hadamard surfaces are quasi-conformal.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Study on surfaces pinched by curvature in space forms converging under specific conditions.

Study pinched submanifolds, proving homology vanishing results.

Paper finds critical metrics with pinched curvature are geodesic balls.

Simon type monotonicity formulas for the Willmore functional $\int | \mathbf{H} |^2$ in the hyperbolic space $\mathbb{H}^n$ and $\mathbb{S}^n$ are obtained. The formula gives a lower bound of $\int_Σ | \mathbf{H} |^2$ where $Σ^2$ is any closed surface in $\mathbb{H}^n$.

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

New restrictions found on 4-manifolds with pinched curvature.

In this paper, the pinching problems of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$.

Classifies self-shrinkers in arbitrary dimensions under specific curvature conditions.

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. We show that if the pinching condion is strong enough with a dependance on the norm of the second foundamental form, then the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

The paper constructs a new metric on Kähler manifolds.

Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…

Proves inequalities for tensor fields on submanifolds using ABP method.

This is a research announcement on an alternative definition of the Casson invariants by means of virtual counting of the moduli space of irreducible representations of the fundamental group into $\SU(2)$. Along the way, by using derived differential geometry, we propose a general framework to obtain invariants from Ch…

We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to characterize some of these submanifolds.

Proves Hamilton's theorem using mean curvature flow.

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…