The moduli continuity method is applied to describe K-polystable log Fano pairs.
problem Describing compact moduli spaces of K-polystable log Fano pairs.
method Applying the moduli continuity method to the log setting.
result Explicit construction of moduli spaces and ampleness of CM line bundle.
In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some o…
In this paper we continue our study on the moduli spaces of flat G-bundles, for any semi-simple Lie group G, over a Riemann surface by using heat kernel and Reidemeister torsion. Formulas for intersection numbers on the moduli spaces over a Riemann surface with several boundary components, over non-orientable Riemann s…
Paper introduces hybrid curves moduli space and canonical measures.
problem Asymptotic geometry of Riemann surfaces and their moduli spaces.
method Definition of hybrid curves and canonical measures.
result Continuous variation of canonical measures over hybrid curves moduli space.
Study continuity of phi-invariant for degenerating graphs.
problem Continuity of phi-invariant for degenerating graphs.
method Use Yuan--Zhang's adelic divisors and follow Yuan's globalization of phi-invariants.
result Asymptotic expression of Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space.
We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pair…
Study special Lagrangian moduli spaces with boundary.
problem Understanding geometric structures on moduli spaces of special Lagrangians.
method Investigates geometric structures and constructs special affine structures and a Hessian metric.
result Constructs a pair of special affine structures and a Hessian metric on the moduli space.
We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that general…
Paper relates new compactification to classical moduli space.
problem Relating new compactification to classical moduli space.
method Defining a continuous finite-to-one surjection between compactifications.
result Uniform upper bound on cardinalities of fibers.
The paper explores metrics on tree moduli spaces and a new topological group.
problem Continuous metrics on tree moduli spaces.
method Proposes and analyzes continuous piecewise-smooth metrics.
result Observes a new abelian topological group structure.
Weil-Petersson volumes vary continuously with weighted points on a projective line.
problem Continuity of Weil-Petersson volumes in moduli space with weighted points.
method Localization and geometric computation methods.
result CM volume converges to geometric volume as weights approach Calabi-Yau geometry.
We extend the Weil-Petersson metric to a projective variety with continuous local potentials.
problem Continuity of the Weil-Petersson potential on moduli spaces of Kähler-Einstein manifolds and varieties.
method Proving the extension of the Weil-Petersson metric as a closed positive current with continuous local potentials.
result The Weil-Petersson metric extends uniquely to the projective variety as a closed positive current with continuous local potentials.
Study of infinite type surfaces' mapping class groups via hyperbolic structures.
problem Understanding mapping class groups of infinite type surfaces.
method Definition of a topology on a moduli space of marked hyperbolic structures.
result Continuous action of mapping class groups on the marked moduli space.
Continuity shown for Yang-Mills flow on semistable bundles.
problem Compactification of moduli space of connections.
method Yang-Mills flow at infinity and comparison of topologies.
result Continuous map defined from Yang-Mills flow to ideal connections.
Study on regularity of exceptional actions and moduli of continuity for circle diffeomorphisms.
problem Regularity of exceptional actions of groups on the circle.
method Analysis of C1,α diffeomorphisms with free orbits and bounded derivatives. result Existence of exceptional C1,α diffeomorphisms under certain conditions on α. Study compactifications of Hermitian-Yang-Mills connections on projective manifolds.
problem Analyzing compactifications of moduli spaces of Hermitian-Yang-Mills connections.
method Defined a gauge theoretic compactification and compared it with sheaf theoretic compactifications.
result Gauge theoretic compactification endowed with complex analytic space structure.
Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.
problem Analyzing moduli spaces of Seiberg-Witten equations on manifolds with boundary.
method General regularity theorem, strong unique continuation principle, and gluing theorem for Dirac operators; smoothness of restriction map.
result Proves moduli spaces are Hilbert manifolds and have semi-infinite-dimensionality properties.
Compactifies moduli spaces of abelian differentials with specific zeroes and poles.
problem Constructing a compactification of moduli spaces of abelian differentials.
method Using a blowup of the incidence variety compactification, defining families of projectivized multi-scale differentials, and performing a real oriented blowup.
result The moduli space of multi-scale differentials is a complex orbifold with normal crossing boundary.
The paper studies moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
problem Understanding moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
method Relating convergence of RCD(0,N)-structures to their lifts, constructing Albanese and soul maps, proving their continuity, and constructing examples.
result Construction of moduli spaces with non-trivial rational homotopy groups.
We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geom…
Researchers describe coordinates for a space of convex RP² structures.
problem Describing the topology of a space of convex real projective structures.
method Explicit coordinates found using quotient of neighborhoods by mapping class groups.
result Explicit coordinates provide simpler proof of homeomorphism to cubic differentials.
Study of symplectic cross-ratios in moduli spaces of line configurations.
problem Understanding geometric and combinatorial properties of line configurations in symplectic spaces.
method Analysis of moduli spaces, geometric and combinatorial problems, isomorphism to quotient spaces of linear difference operators.
result Moduli space of Lagrangian configurations is parametrized by n+1 symplectic cross-ratios, satisfying a remarkable relation.
We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.
We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus ≥2 in Sp(4,R). We prove that this is equivalent to classifying the connected components of the moduli space of representations (which consists of conjugacy classes of red…
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space Mˉ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on Mˉ an…
Study geometric structure of HCMU surfaces with singularities.
problem Understanding the moduli space of HCMU surfaces with specified genus and conical angles.
method Geometric structure analysis using classical football decomposition and data representation.
result Determined the dimension of the moduli space of HCMU surfaces.
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
Extends Teichmueller dynamics to flows on moduli spaces of flat bundles.
problem Tackles ergodic properties of flows on moduli spaces of flat bundles.
method Extends Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces.
result Strong mixing properties observed for flows on deformation spaces when G is compact.
We continue the study of linear families of contact forms on 3-manifolds begun in our paper `Contact geometry and complex surfaces'. The present paper introduces Teichmuller and moduli spaces for so-called taut contact circles. By constructing a developing map for taut contact circles, we show that these geometrically …
Study on moduli spaces of negatively curved metrics on surfaces.
problem Understanding the structure of moduli spaces of uniformly negatively curved metrics on surfaces.
method Construction of locally constant functionals based on geodesic string counts.
result Moduli space of metrics on RimesS1 is disconnected. Paper studies compactifications of Higgs bundles and self-duality equations.
problem Compactification of Hitchin moduli space and Higgs bundles.
method Analyzes maps between algebraic and analytic compactifications.
result Map between compactifications fails to be continuous at boundary over discriminant locus.
Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds
problem Establishing an equivalence between diffeological substacks of Higgs and flat bundles
method Using diffeological moduli stacks
result Shows equivalence of categories between semistable Higgs bundles and flat bundles
Geometric correspondence links flow metrics to reparameterizations.
problem Linking flow metrics to reparameterizations of geodesic flows.
method Analysis of Mineyev's flow space and Green metrics.
result First examples of continuous reparameterizations on negatively curved manifolds.
This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…
In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the p…
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
This is the extended version of the paper "Special Lagrangian conifolds, I: Moduli spaces", which discusses the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. The conif…
The paper provides methods to orient gauge-theoretic moduli spaces.
problem Orienting gauge-theoretic moduli spaces for various applications.
method Develops techniques to prove orientability and construct canonical orientations.
result Constructs canonical orientations on gauge theory moduli spaces.
Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.
problem Constructing local models for vector bundles on Kähler manifolds.
method Applying Geometric Invariant Theory to Kähler manifolds to construct analytic GIT-quotients.
result Existence of Weil-Petersson forms on parameter spaces for stable vector bundles.
Analyzes the moduli space of Higgs bundles to prove its quasi-projectivity.
problem Proving the quasi-projectivity of the moduli space of Higgs bundles.
method Uses analytic methods and the symplectic cut to construct a compactification.
result Proves the quasi-projectivity of the moduli space of Higgs bundles.
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
We study the deformations of twisted harmonic maps f with respect to the representation ρ. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of f in terms of Hodge theory; we apply this result to the moduli space of reductive representations …
We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the b…
In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau …
New flow solves LYZ equation on Kähler manifolds.
problem Solving the LYZ equation on compact Kähler manifolds.
method Introduced a new flow and showed its longtime solution converges to the LYZ equation solution under certain conditions.
result The flow converges to a singular solution on compact Kähler surfaces under specific conditions.
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami …
Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotic…