arXiv research

We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincaré type metrics. We apply this to give new examples of extremal Poincaré type metrics. A key feature is an additional obstruction which has no analogue in the compact case. This condition is conjecturally related to ensuring the metr…

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

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

Characterizes solvability of J-equation on Kähler surfaces with singularities.

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

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on X\D with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to t…

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 paper classifies Poincaré complexes as topological manifolds.

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincaré-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincaré-Hopf index formula to the case of complex vector fi…

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

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

Sharp inequality for compactifying Poincaré-Einstein manifolds.

In this paper, we studied integrals involving both real and complex Hessian operators over bounded domain. Poincare type inequalities were proved in both cases which generalized a early results of Trudinger and Wang.

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist…

Study on 4D Einstein manifolds with Kähler conformal geometry.

We consider two types of minimal Poincaré $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincaré $4$-complexes were introduced by Hambleton, Kreck and Teichner. It…

New Poincaré inequality for differential forms on manifolds.

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

The paper extends stabilization methods to Poincaré Duality complexes.

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 …

4-manifolds with specific groups have unique homotopy types.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

For two complex vector bundles admitting a homomorphism between them, a Poincaré-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincaré-Hopf type f…

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

Study Poincaré inequality in metric spaces via separating sets.

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…

In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.

Paper approximates Kähler metrics with cone singularities near a hypersurface.

We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincaré--Sobolev--Orlicz-type inequalities is discussed.

The paper finds manifold structures on complex spaces.

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain …

The paper examines compactifications of Poincaré-Einstein manifolds and their convergence properties.

Holomorphic families yield metrics with explicit curvature formulas.

In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we inves…

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 studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…

The study establishes inequalities for functions on manifolds using Green function estimates.

Homotopy classification for certain 4-manifolds with dihedral fundamental groups.

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

We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…

In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).

Unique extremal Kähler metric found near a divisor.

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. The…