arXiv research

New integral estimates on substatic manifolds improve Alexandrov Theorem.

The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.

TSAW improves MCMC integral estimation with faster convergence.

Study rigidity on CR Yamabe equation on Sasakian manifolds.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

Study proposes curvature flow model for Drosophila dorsal closure.

In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we …

New Liouville-type results for CR Yamabe equation in Heisenberg group.

Minimal graphs grow slowly on curved spaces, proving constant solutions.

In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow. Moreover, if the initial metric has non-negative bisectional curvature, using Tian…

This paper improves Green's function estimates for compact Kähler manifolds.

Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using randomly generated samples of the function. In light of the increasing interest in uncertainty quantification and robust design applications in aerospace engineering, the calculation of expected values of such functions (e…

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems…

The paper proves growth estimates for subsolutions of quasilinear equations.

We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott. To do this we adapt the technique of quantitative s…

Contextual linear optimization shows naive plug-in methods can outperform direct optimization.

We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theo…

We prove a Davies type double integral estimate for the heat kernel $H(y,t;x,l)$ under the Ricci flow. As a result, we give an affirmative answer to a question proposed by Chow etc.. Moreover, we apply the Davies type estimate to provide a new proof of the Gaussian upper and lower bounds of $H(y,t;x,l)$ which were firs…

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

Study critical metrics on manifolds with boundary using integral and boundary estimates.

In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…

We consider the integrals of $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in space forms $\mathcal{Q}_c^{n+1}$ which generalize volume $(r=0)$, total mean curvature $(r=1)$, total scalar curvature $(r=2)$ and total curvature $(r=n)$. Among other results we prove that a complete properly immersed hypersurfac…

Study extends convexity in curved spaces using fractional integrals.

Optimal Strichartz estimates for Schrödinger on Zoll manifolds.

New findings show ETO outperforms IEO in well-specified models with sufficient data.

Study Kähler-Einstein potentials on stable varieties near singularities

The paper classifies solutions to a specific elliptic equation in the Heisenberg group.

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical with respect to any Killing field, then $F(M^n)$ is a flat disk. This result is ind…

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\logλ)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, Gérard …

Sharp inequalities for weighted log canonical thresholds derived.

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…

We show that for a smooth closed curve $γ$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $e_λ$ and $e_μ$ restricted to $γ$, $|\int e_λ\overline{e_μ}\,ds|$, is bounded by $\min\{λ^\frac12,μ^\frac12\}$. Furthermore, given $0<c<1$, if $0<μ<cλ$, we prove that $\int e_λ\overline{e…

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree c…

Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we …

Let $Φ:TM\to TM$ be a positive-semidefinite symmetric operator of class $C^1$ defined on a complete non-compact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $Φ$ and on the second fundamental form to guarantee that either $Φ\equiv 0$ or the integra…

We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee …

An optimal dynamic treatment regime (DTR) consists of a sequence of decision rules in maximizing long-term benefits, which is applicable for chronic diseases such as HIV infection or cancer. In this paper, we develop a novel angle-based approach to search the optimal DTR under a multicategory treatment framework for su…

Algorithm selects variables and bandwidths for geographically weighted regression.

New method improves curvature estimates for stable surfaces.

The paper proves new rigidity results for critical metrics of quadratic curvature functionals.

New insights into bias-variance tradeoff for data-driven optimization under local misspecification.

The paper establishes curvature estimates for solitons in higher dimensions.

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

Integrates estimation and optimization for uncertain parameters.

Q-NETs use neural networks to estimate integrals of low-dimensional functions efficiently.