Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.
We describe and construct here pseudo-Hermitian structures θ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric ωB locally away from singular set. The paper proves conditions for Kähler manifolds with negative curvature.
problem Conditions for existence of Kähler-Einstein metrics and holomorphic curves.
method Analyzes compact Kähler manifolds homotopic to negatively curved Riemannian manifolds.
result Compact Kähler manifolds with negative curvature admit Kähler-Einstein metrics of general type.
Kähler-Einstein metrics found on compactifications of groups.
problem Existence of Kähler-Einstein metrics on group compactifications.
method Continuity method, real Monge-Ampère equation, invariance under maximal compact subgroup.
result Necessary and sufficient condition for existence of Kähler-Einstein metrics.
In 2D, a conjecture about the Ricci tensor of Kähler-Einstein metrics for convex bodies is verified.
problem Verifying a conjecture about the Ricci tensor of Kähler-Einstein metrics for convex bodies in 2D.
method Analyzing the Kähler-Einstein equation and Hessian metric for convex bodies in 2D.
result The Ricci tensor of the Hessian metric is uniformly bounded by a specific value in 2D.
The study establishes conditions for stratified spaces to satisfy RCD(K, N) curvature-dimension condition.
problem Conditions for stratified spaces to satisfy RCD(K, N) curvature-dimension condition.
method Proves conditions for stratified spaces to satisfy RCD(K, N) using Ricci tensor bounds and cone angles.
result New examples of metric measure spaces satisfying RCD(K, N) curvature-dimension condition.
Defines (p,q) hermitian geometry and formulates it in generalised complex geometry.
problem Defines (p,q) hermitian geometry and formulates it in generalised complex geometry. method Formulates (p,q) hermitian geometry in generalised complex geometry. result Provides explicit formulae for the map to generalised geometry.
Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
A car's motion shows flat parabolic geometry.
problem Understanding the geometry of a car's motion.
method Analyzing a car as a nonholonomic system.
result Shows a car's motion as a flat parabolic geometry.
Generalizes Riemannian geometry concepts to reductive Cartan geometries.
problem Applying Riemannian geometry concepts to a broader class of geometries.
method Defined covariant derivatives and geodesics for reductive Cartan geometries, then proved analogous results.
result A generalization of the Hopf-Rinow theorem with a concise proof.
Hessian geometry explains special geometries in N=2 theories.
problem Explaining special geometries in N=2 theories. method Formulating N=2 theories in terms of Hessian structures. result Hessian geometry relates special geometries of vector multiplets to hypermultiplet geometries.
Abstract: Surveying aspects of complex dimension two anti-canonical pairs.
problem Smooth topology, algebraic geometry, symplectic geometry, and contact geometry of anti-canonical pairs.
method Survey and review of existing work.
result Survey of various geometric properties of anti-canonical pairs.
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Develops theory of Cartan geometries on skeletons and morphisms induced by extension functors.
problem Describing categories of Cartan geometries with additional morphisms.
method Using extension functors to define new categories of Cartan geometries and studying their properties.
result Shows functors between categories of Cartan geometries with morphisms induced by extension functors and categories of Cartan geometries modeled on skeletons.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
Classifies 5D homogeneous geometries with nontrivial reducible linear isotropy.
problem Classifying 5D homogeneous geometries with specific properties.
method Thorough classification using Thurston's criteria and analysis of linear isotropy representations.
result Found a countably infinite family of geometries diffeomorphic to S3imesS2. New definition of Born geometry connects to known geometries.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable geometries.
Surveying recent developments in Hitchin moduli space geometry.
problem Understanding the asymptotic geometry of Hitchin moduli space.
method Introduction to Hitchin moduli space and hyperkähler geometry.
result Recent developments in asymptotic geometry of Hitchin moduli space.
The paper extends group constructions to coset geometries, creating new ways to combine geometries.
problem Combining and gluing incidence geometries in a general framework.
method Extending classical group-theoretic constructions to coset geometries.
result Provides a general framework for combining or gluing incidence geometries.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
Uses Dirac geometry to prove Poisson geometry results.
problem Classical results in Poisson geometry.
method Dirac geometry techniques.
result Proofs of Poisson geometry results.
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
Introduces logarithmic Cartan geometry on complex manifolds with singularities.
problem Holomorphic Cartan geometry with singularities.
method Definition and study of logarithmic Cartan geometry on complex manifolds with polar part supported on a normal crossing divisor.
result Push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Maps strong Kähler with torsion to Generalised Complex Geometry.
problem Representing strong Kähler with torsion in Generalised Complex Geometry.
method Analogy to Gualtieri map for (2,2) models. result Representation of strong Kähler with torsion in Generalised Complex Geometry.
Simplified Beltrami's theorem using geometry.
problem Beltrami's theorem
method Parabolic differential geometry
result Simplified approach to Beltrami's theorem
Talks about new methods in differential geometry.
problem None explicitly stated in the abstract.
method Stresses the neighbor relation as a basic notion.
result Not explicitly stated in the abstract.
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
Constant curvature models in sub-Riemannian geometry are explored.
problem Understanding constant curvature models in sub-Riemannian geometry.
method Cohomological description of principal invariants for parabolic geometries.
result Constant curvature models are discussed for specific sub-Riemannian geometries.
Paper classifies 5D Thurston geometries, finding an uncountable family.
problem Classifying 5-dimensional Thurston geometries.
method Summarizes the general classification, providing a full list and examples.
result An uncountable family of geometries discovered.
The paper extends Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
problem Extending Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
method Extending the Ruh-Vilms theorem to Weitzenböck geometry, considering the flat geometry with torsion.
result The Laplacian of the Gauss map for hypersurfaces in Weitzenböck geometry is related to the mean curvature vector field.
New algebraic-geometry method for Ribaucour transformations.
problem Classical differential geometry problems.
method Algebraic-geometry approach to constructing orthogonal nets.
result Obtains smooth orthogonal nets as Ribaucour transformations.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Paper establishes equivalence in Minkowski spaces with Finsler geometry applications.
problem Equivalence of Minkowski spaces in differential geometry.
method Using centro-affine differential geometry results, the paper establishes an equivalence theorem.
result Characterizations of Berwald spaces in Finsler geometry.
Two open problems in Sasaki geometry are discussed.
problem Open problems in Sasaki geometry.
method Discussion and description of open problems.
result Two open problems in Sasaki geometry are identified.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
The study of special metrics in various parabolic geometries.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.