arXiv research

Study on 3-manifolds finds regular conformal metrics for rough metrics.

In this paper, we study generalized Douglas-Weyl $(α, β)$-metrics. Suppose that an regular $(α, β)$-metric $F$ is not of Randers type. We prove that $F$ is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if $F$ is not a Berwald met…

Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.

Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.

Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.

Defines new metric space sections with Ahlfors-David regularity.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

In this paper, we study almost regular Landsberg general $(α,β)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general $(α,β)$-metrics under the conditon that $β$ is closed and conformal to $α$. Under this condition, we p…

High-dimensional prediction is a challenging problem setting for traditional statistical models. Although regularization improves model performance in high dimensions, it does not sufficiently leverage knowledge on feature importances held by domain experts. As an alternative to standard regularization techniques, we p…

Proves regularity for quasilinear elliptic equations in metric spaces.

Derives spacetime regularity under specific curvature conditions.

New bounds for low-regularity Riemannian metrics defined via distributional curvature.

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M…

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value …

We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This…

Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.

The paper proves the existence of a special type of metric on complex manifolds.

Proves regularity of harmonic maps into Teichmüller space.

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

Study shows how maps from certain geometric spaces behave near their edges.

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…

A gap in the proof of the finsler "unicorn" conjecture in the paper "Regular Landsberg metrics are always Berwald" by Z. I. Szabo is pointed out

We study the regularity properties for solutions of a class of Schrödinger equations $(Δ+ V) u = 0$ on a stratified space $M$ endowed with an iterated edge metric. The focus is on obtaining optimal Hölder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential.

New bounds for geometric flows of Hermitian metrics established.

Low regularity spacetimes split into simpler structures.

Study improves regularity estimates for harmonic maps into ellipsoids.

New static vacuum metrics confirmed for near Euclidean boundary data.

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…

Rapidly growing product lines and services require a finer-granularity forecast that considers geographic locales. However the open question remains, how to assess the quality of a spatio-temporal forecast? In this manuscript we introduce a metric to evaluate spatio-temporal forecasts. This metric is based on an Opti- …

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial res…

Establishes 4D regularity for certain metric spaces.

This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness …

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$…

The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.

We show that #8(S^2 times S^3) admits two 8-dimensional complex families of inequivalent non-regular Sasakian-Einstein structures. These are the first known non-regular Sasakian-Einstein metrics on this 5-manifold.

Characterizes minimizing curves in Riemannian manifolds.

We show that Cheeger deformations regularize $G$--invariant metrics in a very strong sense.

Improved robustness in multivariate regression and classification with DRO under Wasserstein metric.

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

We propose a general information-theoretic approach called Seraph (SEmi-supervised metRic leArning Paradigm with Hyper-sparsity) for metric learning that does not rely upon the manifold assumption. Given the probability parameterized by a Mahalanobis distance, we maximize the entropy of that probability on labeled data…

New class of complex manifolds defined, properties studied.

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|β\|_α>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…

We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the doma…

Introduces Lie group actions in smoothing processes for currents and spaces with curvature.

New method finds smooth isometric immersions for low regularity metrics, achieving full flexibility.

Generic smooth boundaries for isoperimetric regions in 8D manifolds.

We prove a regularity result for unit volume conformal metrics with integral scalar curvature bounds for $p>n/2$ and first eigenvalue of $Δ$ bounded from below by a constant $B > Λ_1(S^n,[g_{st.}]).$

Study shows unique harmonic metrics for certain Higgs bundles over non-compact surfaces.