arXiv research

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Constructs metrics with negative constant scalar curvature.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and negative Euler characteristic has at most negative of the Euler characteristics many small eigenvalues.

Paper extends Willmore inequality to manifolds with negative Ricci curvature.

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We con…

Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space.

New complete hypersurfaces found in affine geometry.

Study the structure of Kähler foliations with negative Ricci curvature.

Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the…

Constructs metrics with negative curvature on specific manifold types.

The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that $π_{1}(\mathcal{T}^{<0}(M))$ is non-trivial. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space…

New rigidity theorems for spin fill-ins with non-negative scalar curvature.

We give examples of harmonic maps between negatively curved manifolds with special properties. These negatively curved manifolds do not have the homotopy type of a locally symmetric space.

Anosov geodesic flow proven in non-compact manifolds with negative curvature.

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.

We introduce two types of ordinal pattern dependence between time series. Positive (resp. negative) ordinal pattern dependence can be seen as a non-paramatric and in particular non-linear counterpart to positive (resp. negative) correlation. We show in an explorative study that both types of this dependence show up in …

New method learns from either positive or negative feedback alone.

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negative…

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.

Study on moduli spaces of negatively curved metrics on surfaces.

We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…

Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.

Study on pairwise counter-monotonicity, a type of negative dependence.

In this paper, we consider a fully nonlinear problem on manifolds with boundaries of negative admissible curvatures. As a consequence, we conclude the existence of certain types of metrics on the general differential manifolds with boundaries.

Classifies contact structures on negative-definite Seifert fibred spaces.

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Li…

The paper models stock returns using $q$-Gaussians and negative binomials.

Given a negatively curved geodesic metric space $M$, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of $M$ in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithme law-type results for the s…

The study confirms a conjecture about Kähler manifolds with quasi-negative curvature.

FTRL algorithm with negative entropy regularizer achieves best-of-three-world results for linear bandits.

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.

The paper examines functional properties on manifolds with very negative curvature.

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

The paper classifies Legendrian and transverse knots in cable knot types.

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji's …

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

Global well-posedness and asymptotic convergence for vacuum Einstein's equations proved.

Extends Bochner's theorem to include small positive Ricci curvature.

We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.

We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds wit…

New black hole solutions with positive and negative masses in 4 and 5 dimensions.