The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
The paper solves the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
problem Solving the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
method Establishes a structure theorem for minimizing sequences, proving the limit of such sequences is identified by a finite collection of isoperimetric regions.
result The limit of a minimizing sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery Ricci tensor. There are tw…
Noncompact RCD spaces with maximal first Betti number are rigid.
problem Characterizing noncompact RCD spaces with maximal first Betti number.
method Analyzing properties of noncompact RCD spaces with maximal first Betti number.
result Spaces with maximal first Betti number are either flat Riemannian manifolds or metric products.
Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.
problem Estimating gradients of solutions to a specific type of partial differential equation.
method Using the Finslerian Allen-Cahn equation as an Euler-Lagrange equation to a Liapunov entropy functional, proving gradient estimates on compact and noncompact Finsler metric measure spaces.
result Global and local gradient estimates of positive solutions to the Finslerian Allen-Cahn equation.
We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition proper…
Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.
problem Establishing gradient estimates for the Finslerian Fisher-KPP equation.
method Global gradient estimates on compact and noncompact Finsler metric measure spaces using the traditional CD(K,N) condition and new comparison theorems. result Global gradient estimates for positive solutions of the Finslerian Fisher-KPP equation.
We consider a complete noncompact smooth metric measure space (Mn,g,e−fdv) and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative f-subharmonic function with bounded weighted L1 norm is constant.
In this short paper we study Lfp-Liouville property with 0<p<1 for nonnegative f-subharmonic functions on a complete noncompact smooth metric measure space (M,g,e−fdv) with Ricfm bounded below for 0<m≤∞. We prove a sharp Lfp-Liouville theorem when 0<m<∞. We also prove an $…
Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition RCD(K,∞) introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is σ-finite; in …
We prove a Liouville property for any f-harmonic function with polynomial growth on a complete noncompact smooth metric measure space (M,g,e−fdv) when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. We construct complete noncompact Riemannian metrics with G2-holonomy on noncompact orbifolds that are R3-bundles with the twistor space Z as a spherical fiber.
Study finds conditions for minimal surfaces in noncompact spaces.
problem Existence of minimal surfaces in noncompact spaces with nonnegative scalar curvature.
method Conditions on a 2-sphere and its boundary.
result Existence of a closed minimal surface homologous to the boundary.
New proof shows no negative curvature Einstein metrics in specific dimensions.
problem Proving nonexistence of certain Einstein metrics in 9 and 10 dimensions.
method Cohomogeneity-one approach to show nonexistence of negative curvature Einstein metrics.
result Noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions.
We study both function theoretic and spectral properties on complete noncompact smooth metric measure space (M,g,e−fdv) with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive f-harmonic functions and obtain as a consequence the strong Liouville property under…
Sharp log-Sobolev inequalities proved for CD(0,N) spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in CD(0,N) spaces. Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
In this paper, using the structures of cone and bicone fields on vector bundles, the author introduces a ILB (inverse limit of Banach)- manifold structure on M the space of Riemannian metrics on a noncompact manifold M. In the last section, it is proven that, this way, on the open submanifold $\mathcal{M}_…
Study characterizes non-collapsed RCD(K, N) spaces using heat kernel metrics.
problem Characterize non-collapsed RCD(K, N) spaces via heat kernel metrics.
method Investigate the second principal term in heat kernel metrics and prove divergence free property.
result Proves non-collapsed property via divergence free property of heat kernel metrics.
Researchers describe horofunctions in noncompact Hermitian symmetric spaces.
problem Understanding horofunctions in noncompact Hermitian symmetric spaces.
method Realized noncompact Hermitian symmetric spaces as open unit balls in Banach spaces with Jordan structures.
result Complete description of horofunctions in the metric compactification.
In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that…
Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply …
Paper solves complex equations on noncompact manifolds.
problem Establishing estimates and existence for fully non-linear equations.
method General class of fully non-linear equations on Kähler and Hermitian manifolds.
result Constructs complete Kähler metrics with prescribed volume forms.
Study gives conditions for noncompact manifolds to have positive scalar curvature.
problem Conditions for noncompact manifolds to have positive scalar curvature.
method Analyzes finite and infinite volume cases to find obstructions.
result Provides conditions for noncompact manifolds to admit metrics with positive scalar curvature.
The paper proves geometric inequalities for hypersurfaces in weighted manifolds.
problem Geometric inequalities for hypersurfaces in weighted manifolds.
method Noncompact smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature.
result Sharp geometric inequalities for the boundary of open sets in weighted manifolds.
We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geomet…
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
The study restricts scalar curvature on certain noncompact manifolds.
problem Ruling out complete metrics with nonnegative scalar curvature.
method Conditions and examples of positive scalar curvature metrics.
result Examples of manifolds with positive scalar curvature.
The noncompact Yamabe flow can lead to incomplete metrics over infinite time.
problem Incompleteness of noncompact Yamabe flow solutions over infinite time.
method Analysis of long-time behavior of the noncompact Yamabe flow.
result Existence of a long-time solution that is complete for each time but converges to an incomplete metric.
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.
problem Equivalence of invariant metrics on noncompact Kähler manifolds with bounded curvature.
method Use boundedness of ratio between Bergman kernel and wedge product of metric in fundamental domain.
result Equivalence of Bergman metric and Kähler-Einstein metric under specific conditions.
In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type extension theorems, and as an application we exhibit the local contractibility of th…
Real flag manifolds are the isotropy orbits of noncompact symmetric spaces G/K. Any such manifold M enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group G, and it is embedded in euclidean space as a taut submanifold. The aim of the paper is to link these two …
We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…
Novel neural network approach on hyperbolic and SPD spaces.
problem Developing neural networks on symmetric spaces of noncompact type.
method Unified formulation of distance from a point to a hyperplane.
result Closed-form expression for point-to-hyperplane distance in higher-rank spaces.
We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If M is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg bo…
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
Extends construction of Kähler-Einstein metrics to noncompact manifolds.
problem Construct complete Kähler-Einstein metrics on noncompact manifolds.
method Iterative construction using Berndtsson's method.
result Induces semipositively curved metric on relative canonical bundle.
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
Classification extended for reducible symmetric spaces.
problem Classifying homogeneous foliations on symmetric spaces.
method Extended classification from irreducible to reducible symmetric spaces.
result Classification completed for all noncompact symmetric spaces.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.