Study classifies special metrics on specific surfaces.
problem Classifying extremal Kähler metrics with singularities.
method Analyzes Hessian of the Curvature of the Metric on K-surfaces.
result Identifies non-CSC HCMU metrics on and .
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Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
New Finsler metrics derived from pedal curves.
problem Constructing new Finsler metrics.
method Using pedal curves or surfaces of other curves/surfaces.
result Generalization of slope metric.
Study connects two types of metrics on complex surfaces.
problem Interrelations between bi-Hermitian and locally conformally Kähler metrics.
method Analysis of metrics on complex surfaces.
result Found interconnections between bi-Hermitian and locally conformally Kähler metrics.
Unique AdS spacetime found with prescribed metric on a convex surface.
problem Finding an AdS spacetime with a specific metric on a convex surface.
method Constructing a quasifuchsian AdS spacetime with a past-convex Cauchy surface.
result Existence and uniqueness of a quasifuchsian AdS spacetime with the specified properties.
Metric surfaces can be divided into small triangles.
problem Decomposing metric surfaces into triangles.
method Proving any metric space homeomorphic to a surface can be divided into non-overlapping convex triangles of small diameter.
result Metric surfaces can be decomposed into triangles of arbitrarily small diameter.
Minimal surfaces help prove a conjecture about special metrics.
problem Proving Arthur L. Besse's conjecture about CPE metrics.
method Using the theory of minimal surfaces.
result The conjecture is proven for 3D manifolds.
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
The study finds conditions for certain surfaces to have a specific type of metric.
problem Understanding the geometry of surfaces with specific metrics.
method Analyzes surfaces of revolution and derives conditions for a strongly convex slope metric.
result Necessary and sufficient conditions for surfaces of revolution to admit a strongly convex slope metric are established.
A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.
Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This com…
Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfacesmath.MG
Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.
problem Embedding flat metrics on hyperbolic surfaces into (2+1)-spacetimes.
method Using convex polyhedral Cauchy surfaces and Teichmüller space properties.
result Existence and uniqueness of flat metrics embedding in (2+1)-spacetimes.
Harmonic metrics of generically regular semisimple Higgs bundles on non-compact Riemann surfacesmath.DG
Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.
problem Proving the existence and uniqueness of harmonic metrics on Higgs bundles.
method Analyzing Higgs bundles equipped with a non-degenerate symmetric pairing on non-compact Riemann surfaces.
result Proving the existence and uniqueness of compatible harmonic metrics under certain conditions.
Study K3 surfaces and their metrics, focusing on dynamics.
problem Understanding dynamics on K3 surfaces.
method Interactions between K3 surface geometry and Ricci-flat metrics, dynamical study of automorphisms.
result Positive entropy automorphisms on K3 surfaces.
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of -invariant Vaisman metrics, analysis of pluriclosed flow behavior.
result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
Flexible metrics found on a genus 2 surface.
problem Identifying non-rigid hyperbolic cone metrics on a genus 2 surface.
method Using a theorem by Erlandsson, Leininger, and Sadanand.
result Nine mapping class group orbits of non-rigid metrics found.
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.
We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and its partial completion can be given by the moduli space of metric graphs for a fat graph associated to the same bordered surface equipped with pressure metric. As a corollary, we show that the pressure metric is not a con…
Survey on uniformization of metric surfaces, including fractal and topological manifolds.
problem Uniformization of metric surfaces homeomorphic to 2D topological manifolds.
method Various uniformization theorems, including quasisymmetric and quasiconformal approaches.
result Uniformization results for metric spheres and arbitrary metric surfaces.
Lipschitz maps on metric surfaces are rigid if they preserve area.
problem Understanding the rigidity of Lipschitz maps on metric surfaces.
method Established a coarea inequality for continuous Sobolev functions on metric surfaces.
result Proved that 1-Lipschitz maps from a closed metric surface to a closed Riemannian surface preserving area are isometries.
Non-convex extremal length found in surface metrics.
problem Extremal length functions on surfaces are not always convex.
method Used harmonic maps to -trees and minimal surfaces in .
result Found measured foliations with non-convex extremal length functions.
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.
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the -class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the -class of the Tricerri/Vaisman metric.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in . In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol3 with a one-param…
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.
Simply connected moduli space of Ricci flat metrics on K3 surfaces.
problem Understanding the topology of Ricci flat metrics on K3 surfaces.
method Analyzing the moduli space of metrics with unit volume.
result The moduli space is simply connected and has cohomology matching the automorphism group.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metricsmath.AG
Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.
problem Characterizing log del Pezzo surfaces with specific geometric properties.
method Examining two classes of non-toric log del Pezzo surfaces and analyzing their geometric properties.
result Examples found that admit Kähler-Ricci solitons but not Sasaki-Einstein cone links.
The paper proves the existence of a special Kähler metric on a minimal ruled surface.
problem Existence of higher extremal Kähler metrics on a minimal ruled surface.
method Proved the existence of a higher extremal Kähler metric by showing it satisfies a specific equation and computed the top Bando-Futaki invariant.
result Higher extremal Kähler metrics exist on a minimal ruled surface, but not higher constant scalar curvature Kähler metrics.
Generalizes Gauss-Bonnet to metrics with logarithmic singularities.
problem Calculating curvature for metrics with singularities on compact surfaces.
method Proves a generalized Gauss-Bonnet formula under Lebesgue integrability condition.
result Establishes formula for special Kähler metrics with meromorphic cubic differentials.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on with two conical singularities.
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging norm of second fundamental form.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
We show that every Kato surface (or surface with a global spherical shell) admits a locally conformally Kaehler metric.
Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.
problem Compare and study surfaces with opposite Gaussian curvatures under two different metrics.
method Consider surfaces in homogeneous 3-manifolds with two metrics, impose extrinsic curvature conditions, and analyze Gaussian curvature functions.
result Identify and characterize anisocurved surfaces with opposite Gaussian curvatures under both metrics.
The study finds points on surfaces where a tensor is conformal to a metric.
problem Existence of conformal points on surfaces.
method Analyzes symmetric bilinear two-tensor fields and Riemannian metrics.
result Provides conditions for the existence of conformal points.
The paper studies geodesics on an icosahedron's Riemann surface.
problem Geodesics on a icosahedron's Riemann surface.
method Constructs a Riemann surface and analyzes geodesics on it.
result Discovers properties of geodesics under a flat metric.
In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing and establishing connections to SKSs and Jacobian elliptic K3 surfaces.
result Established a bijection between integral singular SKSs on and Jacobian elliptic K3 surfaces.
Study on metrics of surfaces with curvature and boundary conditions.
problem Understanding metrics of surfaces with curvature and boundary conditions.
method Revisits and generalizes Ricci-Curbastro's theorem to constant mean curvature surfaces and minimal hypersurfaces.
result New versions of Ricci-Curbastro's theorem for surfaces with curvature and boundary conditions.