arXiv research

Paper finds smooth convex solutions to curvature problem.

Paper proves existence of convex hypersurfaces with prescribed $p$-curvature.

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to …

Finite orbit of representations implies finite image for surface groups.

Proves existence of smooth convex solutions to capillary curvature equations.

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate …

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvat…

Compactness theorem for Riemannian manifolds with volume and curvature bounds.

Study on spaces of metrics with intermediate curvature bounds.

Study calculates the elastic energy of curves on a sphere.

Study on convex capillary hypersurfaces with Lp curvature in half-space.

The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.

We study the horizontally regular curves in the Heisenberg groups $H_n$. We show the fundamental theorem of curves in $H_n$ $(n\geq 2)$ and define the concept of the orders for horizontally regular curves. We also show that the curve $γ$ is of order $k$ if and only if $γ$ lies in $H_k$ but not in $H_{k-1}$ up to a Heis…

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ i…

The paper studies special Finsler spaces with $H_{p}$-scalar curvature.

New geometric system from Hessian operators offers solutions to geometric problems.

The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.

Characterizes limits of Ricci flows and their singularities.

Derives integral formula for differential forms on compact spaces with applications.

New tensors reveal full curvature structure from Riemann tensor.

Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.

Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff d…