arXiv research

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

Derives $L^{\infty}$ estimate for Kähler-Ricci flows with weaker conditions.

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

New proof of Kähler-Einstein Fano manifold $L^\infty$ estimates.

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $\varphi \colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_\varphi\leq \infty$. Our proof holds for the bounded case $r_\varphi< \infty$, recovering …

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

New proof of $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes.

Adapts PDE method to prove $L^\infty$ estimates for complex Hessian equations.

In this work we construct an optimal linear shrinkage estimator for the covariance matrix in high dimensions. The recent results from the random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The developed distribution-free es…

The paper studies the asymptotic behavior of adversarial training under $\ell_\infty$-perturbation.

The paper analyzes kNN density estimation's convergence rates under different conditions.

Note on gradient estimates for complex Monge-Ampere equation.

Estimates log-likelihood of interacting particle systems using virtual particles.

Develops a nonparametric method to estimate isotropic covariance functions efficiently.

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}\colon \mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\rightarrow G$, i.e.…

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimated directly, wit…

Unified estimate for complex Monge-Ampère equations on Kähler manifolds.

On the ambient space of a Lie group with a left invariant metric that is isometric and isomorphic to a semidirect product $\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $Ω\subseteq \mathbb{R}^2\rtimes_A\{0\}$ and vertical $π$-graphs over $Ω$ and study the partial differential equation a function $u:Ω\rightarro…

Let $(X,d,μ)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carré du champ". Assume that $(X,d,μ,\E)$ supports a scale-invariant $L^2$-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\i…

We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the stud…

The paper derives upper bounds on the MLE error for BTL model under general graphs.

Estimating dimension from sparse random geometric graphs.

We study the parabolic flow for generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.

Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α…

Improved bounds for discrete probability distribution estimation under the ℓ∞ norm.

We provide finite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability density, including entropies and divergences. Rather than plugging a consistent density estimate (which requires $k \to \infty$ as the sample size $n \to \in…

The paper estimates gradients on graphs under specific conditions and applies these estimates to heat equations.

Study improves $L^{\infty}$ estimates and extreme value behavior in stochastic differential games.

Let $A$ be a $W^{1,2}$-connection on a principle $\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\|F_A\|_{L^2(M)}\le Λ$. Our main result is the existence of a global section $σ: M\to P$ with finite singularities on $M$ such that the connection form $σ^*A$ satisfies the Coulomb…

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing …

Sharp $L^\infty$ estimates for non-Kähler manifolds' PDEs are derived.

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates to…

Mean curvature flow with uniform bounds on curvature and its gradient

We study the parabolic complex Monge-Ampère type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Ampére type equations.

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $\|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty],$ where $Δ_p$ denotes the $p…

We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline{\lim}_{r\to \infty} r|Rm|<\frac{1}{5}$, then $|Rm|\leq Ce^{-r}$ for some constant $C>0$, improving a result of [36]. For any f…

The paper estimates the measure of nodal sets for solutions to a specific type of Schrödinger equation.

Uniform estimates for Calabi-Yau degenerations proved.

The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

The paper develops a deep neural network estimator for weakly dependent processes with various loss functions.

We discuss pluripotential aspects of the Monge-Ampère equations on compact Hermitian manifolds and prove $L^{\infty}$ estimates for any metric, as well as the existence of weak solutions under an extra assumption.

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

Study potential theory to detect completeness of Finsler manifolds.

Based on a construction due to B. Güneysu and S. Pigola (\textit{Adv. Math.} \textbf{281} (2015), pp.353--393), for each $p \in [1,\infty]$ and $m \in \mathbb{Z}_{\geq 2}$, we exhibit an $m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calderón--Zygmund estimate \begin{equation*} \|\nabla \nabl…

This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" ta…