Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Paper proves almost Schur Lemma on smooth metric measure spaces.
problem Proving a specific lemma on metric measure spaces.
method Proves almost Schur Lemma using closed smooth metric measure spaces.
result Implications of the lemma for X. Cheng's and De Lellis-Topping's results.
Riemannian metrics and Laplacians defined for complex distributions on manifolds.
problem Defining metrics and Laplacians for distributions on manifolds of varying rank.
method Introduced a Riemannian metric and Laplace operator for generalised smooth distributions on manifolds.
result Essentially self-adjoint Laplacian on compact manifolds, hypoellipticity proven.
Researchers develop weighted GJMS operators for smooth metric measure spaces.
problem Developing mathematical tools for smooth metric measure spaces.
method Constructing and proving formal self-adjointness of weighted GJMS operators.
result Formal self-adjointness of weighted GJMS operators proved.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Introduces Lie group actions in smoothing processes for currents and spaces with curvature.
problem Regularization of currents and metrics on manifolds and spaces with curvature.
method Actions of compact Lie groups in De Rham approximation and smoothing of Riemannian metrics.
result Effective smoothing processes for currents and metrics on manifolds and spaces with curvature.
The paper constructs manifolds without smooth psc metrics but with L∞-metrics that are psc outside singular points.
problem Constructing manifolds without smooth positive scalar curvature metrics.
method Constructing manifolds with point singularities and L∞-metrics that are psc outside the singular set. result Examples of manifolds with point singularities that do not admit smooth psc metrics but do admit L∞-metrics that are psc outside the singular set. Geometric condition ensures smoothness of gravity metrics at shock waves.
problem Ensuring smoothness of gravitational metrics at shock waves in GR.
method Introducing Riemann-flat condition to determine metric smoothness.
result Locally inertial frames always exist for spherically symmetric spacetimes.
Heat kernels exist and are Hölder for rough metrics on smooth manifolds.
problem Existence and regularity of heat kernels on rough metrics.
method Local parabolic Harnack estimates for weak solutions in weighted Sobolev spaces.
result Globally continuous heat kernels are Hölder continuous locally.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Smooth metrics satisfying Penrose inequality are necessarily smooth.
problem Rigidity of Penrose inequality with singular metrics.
method Showed suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality are smooth in specified coordinates.
result Smooth metrics satisfying Penrose inequality are necessarily smooth.
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Study Ricci tensor on spaces with boundary, offering new curvature conditions.
problem Curvature conditions on smooth metric measure spaces with boundaries.
method Generalization of Bakry-Emery's Ricci tensor to spaces with boundaries.
result New approach to curvature-dimension conditions on spaces with boundaries.
Study on curved metrics on manifolds using smoothing theory.
problem Understanding rational homotopy groups of curved metrics on high-dimensional manifolds.
method Classical results in smoothing theory applied to high-dimensional manifolds.
result Smooth M-bundles with fiberwise negatively curved metrics represent elements of finite order in homotopy groups.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. Paper discusses smoothness conditions for metrics on cohomogeneity one manifolds.
problem Smoothness conditions for metrics on cohomogeneity one manifolds.
method Describes metrics in terms of functions along geodesic normal to hypersurfaces, and presents a method to compute these conditions.
result Makes computation of smoothness conditions straightforward.
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.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
Establishes smooth Ricci flows from convex surfaces in 3D space.
problem Existence and uniqueness of Ricci flow starting from convex surfaces.
method Smooth Ricci flows starting from smooth convex surfaces.
result Uniform convergence of metrics to initial convex surface.
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.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
Study confirms conjecture about Kähler metrics on smooth minimal models.
problem Behavior of constant scalar curvature Kähler metrics on smooth minimal models.
method Analysis of metrics in a neighborhood of the canonical class.
result Convergence to singular Kähler Einstein metric in the canonical class.
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in W2,2. result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
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…
Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
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 show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7admitsasmoothstructurewhichhasanEinsteinmetricofscalarcurvatures > 0,asmoothstructurewhichhasanEinsteinmetricwiths < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics.…
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
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 R4. Similarly, a smooth 4-manifold homeomorphic to the produc…
Gradient estimates for nonlinear parabolic equations on metric spaces.
problem Nonexistence of positive solutions to nonlinear parabolic equations.
method Local elliptic and parabolic gradient estimates.
result Conditions guaranteeing the nonexistence of nontrivial positive solutions.
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 Gμ(α,β)=C1(μ(M))∫Mμαμβμ+C2(μ(M))∫Mα⋅∫Mβ for some smoo…
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
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 F-limits.
problem Smooth convergence of F-limit flows. method Extensively studied metric flows and F-limits, showing smooth convergence at regular points. result Each regular point on the limit is a point of smooth convergence.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
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…
Study shows curvature stability under smooth metric convergence.
problem Stability of nonnegative isotropic curvature under metric deformations.
method Introduced method by R. Bamler to study scalar curvature behavior.
result Proved that if metrics converge in C0 norm, resulting metric has isotropic curvature bounded from below.
For smooth metric measure spaces (M,g,e−fdvol) 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…
The paper examines the smoothness of hyperbolic metrics near boundaries.
problem Analyzing the regularity of asymptotically hyperbolic metrics near boundaries.
method Following Michael Anderson's method, the paper studies Cm,α conformally compact Riemannian metrics with Einstein equation. result The conformal compactifications of these metrics are Cm+2,α up to the boundary when Weyl curvature is in Cm,α and the boundary metric is in Cm+2,α. 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.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
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…
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…
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
New Einstein metrics found on manifolds with opposite curvature signs.
problem Finding Einstein metrics with opposite curvature signs on manifolds.
method Reviewing and extending previous work on high-dimensional smooth closed manifolds.
result Proved various related results, including new Einstein metrics.