arXiv research

The study preserves upper bounds of total scalar curvature in conformal classes.

We consider the moduli space of the extremal Kähler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities…

The paper examines sequences of metric spaces converging to compact limits with specific properties.

We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the…

Paper compares total quotient curvature and proves bounds for Einstein metric.

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…

The study constructs bundles with non-multiplicative A-genus and finds non-trivial homotopy groups in spaces of metrics.

Study Kähler geometry on vector bundles over elliptic curves.

The study preserves lower bounds of total scalar curvature under specific metric convergence.

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

New bounds use IPMs to improve generalization in machine learning.

Upper bound for total mean curvature of spin fill-ins is proven.

Optimizes Dirac eigenvalue bound using curvature and quasi-spherical metrics.

Let (X,d) be a metric space and m\in X. Suppose that φ:X\times X\to\mathbold{R} is a nonnegative symmetric function. We define a metric d^{φ,m} on X which is equivalent to d. If d^{φ,m} is totally bounded, its completion is a compactification of (X,d). As examples, we construct two compactifications of (\mathhbold{R}^s…

We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Grom…

The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.

Study bounds total mean curvature of fill-ins with scalar curvature constraints.

The paper finds many negatively curved Kähler metrics on complex manifolds.

Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.

Establishes a Penrose-type inequality for static spacetimes.

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…

Totally geodesic subvarieties in moduli space are locally rigid.

We construct a new Riemannian metric on Goldman space $\mathcal{B}(S)$, the space of the equivalence classes of convex projective structures on the surface $S$, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichm$\ddot{u}$ller space, embe…

New extremal metrics found on Kähler manifolds.

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …

Totally geodesic subvarieties in moduli spaces are studied.

On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvat…

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $σ$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area …

Defines curvature for metric triples in metric spaces.

We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equ…

The universal Teichmüller space is an infinitely dimensional generalization of the classical Teichmüller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature…

Proves sufficiency of countable test plans for BV functions on metric spaces.

KeRNS tackles non-stationary reinforcement learning in metric spaces.

Study on umbilical submanifolds in specific geometric spaces.

New rigidity result for fat bundles with equal vertical curvatures.

On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…

The aim of this short article is to investigate the possibility of existence of totally umbilical isometric immersions in R^3 with isothermal parametrization and harmonic metric.

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

Improved learning rates with new smoothness measure.

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms …

The paper improves reinforcement learning by estimating return distributions efficiently.

Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This…

New Lipschitz bound for ReLU networks resists weight rescaling.

Proves properties of 4-manifolds with scalar curvature constraints.

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

A canonical hyperkaehler metric on the total space $T^*M$ of a cotangent bundle to a complex manifold $M$ has been constructed recently by the author (see alg-geom/9710026). This paper presents the results of alg-geom/9710026 in a streamlined and simplified form. The only new result is an explicit formula obtained for …

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…