arXiv research

Paper extends rigidity and vanishing results for totally real submanifolds under $L^p$-integrable conditions.

Calabi flow extended with bounded curvature integrals.

The paper approximates financial derivatives using neural networks and iterated integrals.

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Ω$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…

In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p

This paper improves the approximation of machine learning models by transforming them to better fit locally $p$-integrable functions.

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-var…

Let $(M, \om)$ be a symplectic manifold, endowed with a compatible almost complex structure J and the associated metric g . For any p \in {1, 2, ... (dim M)/2} the form $\Om := \frac{\om^p}{p!}$ is a calibration. More generally, dropping the closedness assumption on $\om$, we get an almost hermitian manifold $(M, \om, …

Universal approximation for stochastic processes using Brownian motion.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

Unified rigidity theorem for cyclic and alternating surfaces.

Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.

The paper studies surfaces in a bounded domain with orthogonal boundaries and proves curvature estimates.

Develops a theory for mth order p-affine capacity for convex bodies containing the origin.

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough,…

Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear is…

Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.

We prove the exponential law $\mathcal A(E \times F, G) \cong \mathcal A(E,\mathcal A(F,G))$ (bornological isomorphism) for the following classes $\mathcal A$ of test functions: $\mathcal B$ (globally bounded derivatives), $W^{\infty,p}$ (globally $p$-integrable derivatives), $\mathcal S$ (Schwartz space), $\mathcal D$…

Let $\mathcal{C}(\mathcal{R},n,p,Λ,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\leq D$, non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvature condition $\|\mathcal{R}\|_{L^p}\leq Λ$ for some $p>\frac n2$. Let $(M,g_0)$ be a compact Riemannian manifold …

In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ …