arXiv research

Optimal Poincaré constant estimates on manifolds with ends.

Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.

Uniform Poincaré inequalities established for various metric spaces.

Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a con…

The article proves Randers Poincaré disc satisfies isoperimetric equality.

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neuma…

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component of the divisor that the scalar curvature of Poincare type metric outside of D is …

Paper approximates Kähler metrics with cone singularities near a hypersurface.

Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper …

CMS formulation solves Poincare conjecture for all dimensions.

Let $X$ be a compact Kähler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature Kähler metrics on non-compact Kähler manifold $X-S$ with Poincaré--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and …

Efficient algorithm for sampling from arbitrary compact bodies.

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective struc…

Proves inequality linking function deviation to gradient norm on compact manifolds.

Paper proves existence of a CMC hypertorus in 4D sphere using numerical methods.

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …

The paper proves optimizability implies inequalities for sampling.

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

The paper defines and analyzes Kähler metrics near a compact manifold, showing their deviation from Poincaré-type metrics.

The paper studies inequalities for fractional GJMS operators on conformal infinity.

We prove a sharp Poincaré inequality for subsets $Ω$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument…

In this paper, we mainly study the scattering operators for the Poincaré-Einstein manifolds. Those operators give the fractional GJMS operators $P_{2γ}$ for the conformal infinity. If a Poincaré-Einstein manifolds $(X^{n+1}, g_+)$ is locally conformally flat and there exists an representative $g$ for the conformal infi…

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

Study on functional inequalities on simple edge spaces.

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

Study the spectrum of Poincaré operator in triaxial ellipsoids.

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive …

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincaré-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses Fenchel's theorem about the total curvature of closed space curves in an essential way. Nex…

New trade-off found between accuracy and adversarial robustness in regression.

We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scal…

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation $Δu - e^{2u} = K_0(z)$ on general open surfaces. A few oth…

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

Let $Σ$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$\mathrm{\acute{e}}$ inequality. We show that any positive minimal graphic function on $Σ$ is a constant.

The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riem…

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

Let $(M,\,g)$ be a Poincar$\acute{\text{e}}$-Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant $Q$-curvature in the conformal class of an asymptotically hyperbolic metric close enough to $g$. These metrics are paramet…

In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially…

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

New tensors help determine if metrics are related to Poincaré-Einstein ones.

Let $(M,g)$ be a compact Kähler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré-…