Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.
problem Solving complex Hessian equations with specific singularity types on compact Kähler manifolds.
method Analyzes the total mass of complex Hessian measures and solves equations with prescribed singularities.
result Proves non-decreasing total mass of complex Hessian measures and solves complex Hessian equations.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.
Study finds solutions to Yamabe equation with specific behavior near singular points.
problem Existence of solutions with prescribed asymptotic behavior near singular points of the Yamabe equation.
method Analysis of positive solutions with isolated singularities and asymptotic expansions.
result Existence of solutions with arbitrarily high order of approximation near singular points.
We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.
Paper solves curvature assignment on surfaces with sharp points and edges.
problem Prescribing curvatures on surfaces with conical singularities and corners.
method New variational formulation for surfaces with singularities.
result First results for prescribed curvatures on surfaces with singularities.
Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
Determines conditions for abelian differentials with specific singularities.
problem Finding conditions for abelian differentials with prescribed singularities.
method Analyzes representations of surface fundamental groups and their holonomy.
result Necessary and sufficient conditions for the existence of translation surfaces with prescribed singularities.
Study prescribed scalar curvature on orbifolds with isolated singularities.
problem Prescribed scalar curvature problem on orbifolds with singularities.
method Compactness theorem in dimension 4, existence theorem in dimensions n ≥ 4, use of Leray-Schauder degree.
result Existence of solutions in dimensions n ≥ 4, compactness in dimension 4.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.
Study of Kähler metrics with prescribed singularities on complex spaces.
problem Define and study spaces of Kähler potentials with prescribed singularities.
method Define non-pluripolar products, spaces of finite energy Kähler potentials, and metrics.
result Spaces of finite energy Kähler potentials with prescribed singularities are complete metric spaces.
Survey on metrics with conic singularities on Riemann surfaces.
problem Conformal metrics with conic singularities on Riemann surfaces.
method Study of metrics with constant positive curvature and conic singularities.
result Simplified proofs for interesting cases, including those by C. L. Chai, C. S Lin, and C. L. Wang.
We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with…
We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our res…
Study on continuity of solutions for complex Monge-Ampère equations with movable singularities.
problem Continuity of solutions with prescribed singularities for complex Monge-Ampère equations.
method Strong continuity methods with movable singularities, including Kähler-Einstein metrics.
result Sufficient conditions for strong continuity of solutions and openness results for Fano type equations.
New approach to prescribing Gaussian curvature on spheres with conical singularities.
problem Prescribing Gaussian curvature on the 2-sphere with conical singularities.
method Variational methods not relying on Moser-Trudinger inequality, plus precompactness theorem.
result Sufficient conditions for a positive function to be the Gaussian curvature of a conformal conical metric.
The paper extends entropy concepts to Monge-Ampère measures with prescribed singularities.
problem Investigating entropy for Monge-Ampère measures with specific singularities.
method Generalizing entropy for potentials, studying stability under blow-ups and perturbations, proving Moser-Trudinger inequalities.
result Functions with finite entropy belong to a specific energy class and maintain singularities of the model potential.
Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\smΣ\to\H^{k+1}_\C$ into the (k+1)-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed sing…
The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.
problem Existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles.
method Using Cartan's highest weight theory, the paper establishes an algebraic criterion for topological splitting and decouples the prescribed mean curvature equation.
result A sufficient algebraic condition for realizing an L2-function as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle. In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is …
Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary conditions. We prove existence and uniqueness of singular monopoles satisfying prescribed boundary conditions for any given He…
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic p≥2. In this cas…
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
Constructs flows with prescribed tangent and asymptotic behavior.
problem Creating mean curvature flows with specific tangent flows and asymptotics.
method General theorem allowing construction of flows with prescribed tangent and asymptotic behavior.
result Constructs mean curvature flows with prescribed tangent and asymptotic behavior.
Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic …
We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.
Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
Constructs scalar-flat Kähler metrics with varying conical singularities.
problem Creating scalar-flat Kähler metrics with specific singularities.
method Using LeBrun's ansatz, constructs metrics with varying conical singularities.
result Constructs complete scalar-flat Kähler metrics with prescribed conical singularities.
Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
Research examines curves of degree 8 with specific singularities.
problem Existence of curves with prescribed singularities.
method Algebraic and symplectic approaches.
result Characterization of curves with specific singularities.
Constructs real algebraic functions with specified preimages.
problem Reconstructing smooth functions with prescribed preimages.
method Using real algebraic functions and techniques from singularity theory and differential topology.
result Constructs examples of real algebraic functions with specified preimages.
Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.
problem Prescribing scalar curvature and mean curvature on compact manifolds with boundary.
method Introducing singular metrics inspired by previous work on closed manifolds, proving rigidity results for flat manifolds with totally geodesic boundary.
result Generic scalar-flat manifolds with minimal boundary can have scalar curvature and mean curvature prescribed simultaneously.
Maxfaces can have cuspidal edges near certain singularities.
problem Characterizing singularities on maxfaces.
method Analyzing singular Björling data and proving geometric properties.
result Near a maxface with a specific type of singularity, there exists another maxface with a cuspidal edge.
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n−3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a n…
Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps…
A necessary and sufficient condition for the existence and uniqueness of a conformal metric on 2-sphere of constant curvature 1 and with three conical singularities of prescribed order is given.
In this paper we extend Y.Eliashberg's h-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity ΣI at each point…
For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the com…
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant o…
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…
Singular monopoles are nonabelian monopoles with prescribed Dirac-type singularities. All of them are delivered by the Nahm's construction. In practice, however, the effectiveness of the latter is limited to the cases of one or two singularities. We present an alternative construction of singular monopoles formulated i…
Study Hessian equations on compact Kähler manifolds with prescribed singularities.
problem Characterize finite energy ranges of the Hessian operator and solutions of degenerate complex Hessian equations.
method Reformulate pluripotential results to Hessian setting and use a new method.
result Prove solutions of degenerate complex Hessian equations have the same singularity type as the model potential.
Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.