arXiv research

Proves existence and uniqueness of weighted metrics for smooth spaces.

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

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

The paper constructs manifolds without smooth psc metrics but with $\mathrm{L}^\infty$-metrics that are psc outside singular points.

Smooths metrics with nonnegative scalar curvature near singular sets.

Smooth metrics satisfying Penrose inequality are necessarily smooth.

Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.

Injectivity of X-ray transform proven for non-smooth metrics.

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

Establishes smooth Ricci flows from convex surfaces in 3D space.

This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor m…

Smooth orbit equivalence proves metric equivalence for geodesic flows.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

Study confirms conjecture about Kähler metrics on smooth minimal models.

We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metri…

The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Lapla…

Smooth Kahler-Einstein metrics have been studied for the past 80 years. More recently, singular Kahler-Einstein metrics have emerged as objects of intrinsic interest, both in differential and algebraic geometry, as well as a powerful tool in better understanding their smooth counterparts. This article is mostly a surve…

We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4-manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4-manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scal…

We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic…

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the es…

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$admits a smooth structure which has an Einstein metric of scalar curvature$s > 0$, a smooth structure which has an Einstein metric with$s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics.…

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a new approach to study curvature-dimension condition of smooth metric measure spac…

Proves existence of smooth metrics with specific curvature properties.

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the produc…

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $G_μ(α,β)=C_1(μ(M)) \int_M \fracαμ\fracβμ\,μ+ C_2(μ(M)) \int_Mα\cdot \int_Mβ$ for some smoo…

Gauduchon's theorem extended to singular spaces with smoothing.

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Émery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Ei…

Study smooth convergence of metric flows from $\mathbb{F}$-limits.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spac…

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such t…

Smooth bundles with rough data maintain Hodge kernel isomorphism.

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $ν$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $κ$-noncollapsing property. Finally, we us…

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozek…

New Einstein metrics found on manifolds with opposite curvature signs.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

Metrics are semipositively curved if they meet a specific asymptotic condition.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estima…

Continuous metrics on manifolds with singularities are shown to be Einstein.

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices…

This note is a continuation of the author's paper \cite{Li}. We prove that if the metric $g$ of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constan…