New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
We formulate a quantization commutes with reduction principle in the setting where the Lie group G, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and a…
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
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.
Cohomogeneity-one actions on symmetric spaces of mixed type
problem Classifying cohomogeneity-one actions on symmetric spaces
method Using a new family of diagonal cohomogeneity-one actions
result Reducing the classification problem to symmetric spaces of a single type
Existence proof of noncompact self-shrinkers with arbitrary genus.
problem Existence of noncompact self-shrinkers with arbitrary genus.
method Employing min-max techniques to rigorously prove existence.
result Confirmation of one asymptotically conical end for large genus self-shrinkers.
Morse theory extended to noncompact manifolds with complex geometric data.
problem Extending Morse theory to noncompact manifolds with intricate geometric and homotopy data.
method Defining Morse homology for pairs of manifolds and related geometric/homotopy data, constructing a homotopy coherent diagram of linear maps, and showing it computes Morse homology.
result Morse homology can be computed using a chain complex derived from a homotopy coherent diagram.
Every noncompact surface can be a leaf in a minimal foliation of a 3-manifold.
problem Tackling the minimal foliation structure of noncompact surfaces.
method Using continuous minimal actions of surface groups on the circle to construct minimal foliations.
result Any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3-manifold.
We construct in an explicit algebraic form a family of complete noncompact Ricci-flat metrics which generalize Calabi metrics in real dimension 4(n+1) and with holonomy SU(2(n+1)).
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.
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
New method constructs potential functions for Kähler-Einstein metrics.
problem Constructing potential functions for Kähler-Einstein metrics on pseudoconvex domains.
method Method of potential scaling.
result Existence of 1-parameter family of automorphisms for certain pseudoconvex domains.
Finite time for Ricci flow on certain manifolds.
problem Finite time of Ricci flow on specific manifolds.
method Analysis of homogeneous Ricci flows on noncompact manifolds.
result Finite extinction time for a family of homogeneous Ricci flows.
The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.
problem Analyzing noncompact steady gradient Ricci solitons with nonnegative curvature operator.
method Examining the asymptotic behavior of noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative curvature operator away from a compact set.
result 4D noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling.
Researchers found solutions to a complex equation on spheres, overcoming a key difficulty.
problem Finding solutions to a specific equation on spheres with a background metric.
method Constructed a smooth metric invariant under antipodal map, used a noncompact family of solutions, and addressed the loss of ellipticity.
result Provided solutions to the σ2-Yamabe equation for n=27 and beyond, overcoming a main difficulty. We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite …
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…
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
We survey the known existence and non-existence results for G2-instantons on non-compact cohomogeneity-1 G2-manifolds and their consequences, including an explicit example of a family of G2-instantons where bubbling, removable singularities and conservation of energy phenomena occur. We also describe several o…
Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.
problem Characterizing surfaces with minimal hyperbolic foliations.
method Analyzing codimension one foliations on closed 3-manifolds.
result Noncompact surfaces satisfying a specific condition are homeomorphic to the leaf of a minimal foliation with non-simply connected generic leaf.
Researchers create a family of solitons connecting a cigar to a sphere.
problem Classifying steady gradient Ricci-Yang-Mills solitons on surfaces.
method Constructed a 1-parameter family of solitons on surfaces.
result Any complete steady gradient Ricci-Yang-Mills soliton on a surface must come from this family.
Every noncompact surface has a 3-rigid triangulation.
problem Classifying noncompact surfaces and proving their rigidity.
method Triangulation and minimally rigid structures.
result Every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid.
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of 38π2, i.e., they attain all possible volumes of c…
In [4], we proved that every noncompact ancient κ-solution to the Ricci flow in dimension 3 is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that the same method implies that compact ancient κ-solutions are rotationally symmetric. …
Extends Llarull's theorem to noncompact manifolds with boundary.
problem Generalizing Llarull's theorem to noncompact manifolds with boundary.
method Extends previous results to include compact boundaries.
result Generalized theorem to noncompact manifolds with compact boundaries.
Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.
problem Prescribing Chern scalar curvatures on noncompact manifolds.
method Solving a Kazdan-Warner type equation on noncompact non-Kähler manifolds with an analytic condition.
result Established existence results and provided a new proof of multiplicity theorem.
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.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
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 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 Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci provided examples showing that in the noncompact case the conjecture is wrong. H…
Noncompact Ricci-flat solutions have infinite unstable dimensions.
problem Understanding unstable dimensions of noncompact Ricci-flat solutions.
method Derived sufficient conditions for infinite-dimensional unstable manifolds.
result Noncompact Ricci-flat solutions have uncountably many unstable perturbations.
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.
The paper proves properties of geometric flows on noncompact manifolds.
problem Existence criteria for geometric flows on noncompact affine Riemannian manifolds.
method Obtained existence criteria through a geometric flow on noncompact affine Riemannian manifolds.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature and bounded geometry are diffeomorphic to \(\mathbb{R}^n\) if their tangent bundle has maximal volume growth.
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…
The paper proves uniqueness of Ricci flow on noncompact manifolds.
problem Proving uniqueness of Ricci flow on noncompact manifolds.
method Pseudolocality theorem for L-complete Ricci flow. result Strong uniqueness theorem for L-complete Ricci flow on Euclidean space. Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
problem Noncompact cases of Gauss Curvature Flow on revolution surfaces.
method Examines two noncompact cases of Gauss Curvature Flow on revolution surfaces.
result Examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
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.
Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…
Complete classification of symmetric spaces actions.
problem Classifying symmetric spaces actions.
method Isometric cohomogeneity-one actions classification.
result Classification completed for all symmetric spaces of noncompact type.
Paper proves a noncompact version of Gromov's band-width estimate.
problem Proving a precise upper bound for noncompact Riemannian bands.
method Developed a quantitative partitioned manifold index theory.
result Proved a version of Gromov's band-width estimate for noncompact Riemannian bands.
Study solves Ricci curvature problem for specific noncompact spaces.
problem Solving the Prescribed Ricci Curvature problem for noncompact spaces with two isotropy summands.
method Classified and solved for all simply connected, noncompact G/H with semi-simple G and connected H having two irreducible summands. result Provided solutions to the Prescribed Ricci Curvature problem for all such spaces.
We consider a complete noncompact smooth Riemannian manifold M with a weighted measure and the associated drifting Laplacian. We demonstrate that whenever the q-Bakry-Émery Ricci tensor on M is bounded below, then we can obtain an upper bound estimate for the heat kernel of the drifting Laplacian from the upper b…
New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.
problem Identifying manifolds with noncompact boundaries that cannot be pseudo-collarable.
method Provided examples of Z-compactifiable manifolds with noncompact boundaries that fail to be pseudo-collarable. result Found manifolds with noncompact boundaries that cannot be pseudo-collarable.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. Study extends convergence results to noncompact Ricci flows.
problem Convergence of noncompact nonsingular solutions of the Ricci flow.
method Extends convergence results from compact to noncompact cases.
result Blow up limit of finite time singularity is a gradient shrinking soliton.
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.
Study of invariant solutions for certain PDEs on Riemannian manifolds.
problem Solving PDEs with group invariance on unbounded domains.
method Reduction of unbounded domains to bounded ones using group actions.
result Presentation of a method for studying invariant solutions.