New rigidity results for hypersurfaces in warped product manifolds and Euclidean space.
problem Rigidity of hypersurfaces in warped product manifolds and Euclidean space.
method Weighted Hsiung-Minkowski integral formulas and Brendle's inequality.
result New rigidity results for hypersurfaces in specific manifolds.
New method characterizes minimal surfaces in 3D space.
problem Characterizing minimal surfaces in 3D space.
method Alexandrov Reflection Method
result Embedded minimal free boundary annuli in B3 are the critical catenoid. In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded …
Characterizes orbifolds with upper curvature bounds as reflectofolds.
problem Understanding orbifolds with upper curvature bounds.
method Characterization through Alexandrov curvature and reflectofolds.
result Quotients of Riemannian manifolds by isometries have locally bounded curvature if and only if they are reflectofolds.
New proof of Kähler-Einstein Fano manifold L∞ estimates.
problem Uniform L∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds. method Using Chen-Cheng's auxiliary Monge-Ampère equation and the Alexandrov-Bakelman-Pucci maximum principle, without pluripotential theory.
result Uniform L∞ estimates derived for Kähler-Ricci flow on Kähler-Einstein Fano manifolds. Reflection principle for minimal surfaces in smooth 3-manifolds proved.
problem Understanding minimal surfaces in non-analytic 3-manifolds.
method Reflection principle proved for minimal surfaces in smooth 3-manifolds.
result Explicit application in smooth 3-manifolds.
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…
Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. Proof of Schwarz principle for specific minimal surfaces.
problem Reflection principle for Jenkins-Serrin type minimal surfaces.
method Proof in homogeneous three-manifolds E(κ,τ) for κ≤0 and τ≥0. result Verification of Schwarz reflection principle for new class of minimal surfaces.
Reflection principles for harmonic and holomorphic maps from Hermitian symmetric spaces.
problem Establishing reflection principles for harmonic and holomorphic maps between Riemannian manifolds.
method Introducing recursive real forms and proving reflection principles for harmonic and holomorphic maps from Hermitian symmetric spaces.
result Reflection principles for harmonic and holomorphic maps from a class of Hermitian symmetric spaces.
We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in R3 of constant mean curvature which meet planes Π1 and Π2 in constant contact angles γ1 and γ2 and bound, together with those planes, a…
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.
problem Studying spaces with non-Riemannian curvature beyond Alexandrov geometry.
method Introducing a weak quadruple comparison principle and developing a strainer theory.
result Spaces have constant integer dimension, measure contraction property, and unique Banach tangent cones.
The paper explores reflection principles for lightlike line segments on maximal surfaces.
problem Reflection property does not hold for lightlike line segments on maximal surfaces.
method Analyzes reflection properties for lightlike line segments connecting shrinking singularities.
result Shows a kind of reflection principle for lightlike line segments on maximal surfaces.
Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.
problem Investigating reflection principles for zero mean curvature surfaces in isotropic 3-space.
method Analyzes reflection principles for zero mean curvature surfaces in I3. result Shows a reflection principle for isotropic line segments on zero mean curvature surfaces in I3. Minimal surfaces reflect across spheres, proving annulus uniqueness.
problem Uniqueness of free boundary minimal annuli in balls.
method Reflection principle applied to minimal surfaces meeting spheres at 90 degrees.
result Every embedded free boundary minimal annulus in a ball is the critical catenoid.
We consider the evolution of hypersurfaces on the unit sphere Sn+1 by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such a…
The paper studies large deviation principles for stochastic volatility models with reflection, focusing on binary barrier options and call prices.
problem Large deviation principles for stochastic volatility models with reflection.
method Sample path and small-noise large deviation principles for the log-price process.
result Asymptotic behavior of binary barrier options and call prices in the small-noise regime.
On a multi-assets Black-Scholes economy, we introduce a class of barrier options. In this model we apply a generalized reflection principle in a context of the finite reflection group acting on a Euclidean space to give a valuation formula and the semi-static hedge.
The paper calculates prices for multi-step barrier options under the Black-Scholes model.
problem Calculating prices for multi-step barrier options with varying barriers and time steps.
method Derives a general, explicit expression for option prices using the Black-Scholes model and a multi-step reflection principle.
result Derives a multi-step reflection principle that generalizes the reflection principle of Brownian motion.
In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is …
Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.
problem Anisotropic obstacle problem for minimal surfaces.
method Cahn-Hoffman transform to convert to isotropic problem with generalized Robin boundary condition.
result Optimal regularity of the solution and C1,1 regularity of the free boundary. New stability estimate for spherical symmetry of hypersurfaces with constant mean curvature.
problem Stability of hypersurfaces with constant mean curvature near a sphere.
method Stability estimate using integral identities and inequalities related to torsional rigidity.
result Hypersurfaces can be contained in a spherical annulus with a specific radius difference.
The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
Gradient estimate proved for Donaldson's equation on Kähler manifolds.
problem Proving gradient estimates for Donaldson's equation on compact Kähler manifolds.
method Using uniform upper bounds for trωχφ and Alexandrov-Bakelman-Pucci (ABP) maximum principle. result Gradient estimate for Donaldson's equation derived from uniform bounds.
The paper creates symmetrical discrete minimal nets using Schwarz reflection.
problem Creating symmetrical discrete minimal nets.
method Extending Schwarz reflection principle to discrete minimal surfaces.
result Global examples of discrete minimal nets with high symmetry.
The study examines special domains in S^2 supporting specific solutions to a PDE.
problem Identifying special domains in S^2 supporting positive solutions to a PDE.
method Extends moving plane method and Alexandrov reflection method to prove symmetry.
result Domains must be rotationally symmetric under specific conditions.
We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…
We solve the partial data Calderón problem for the connection Laplacian on Riemann surfaces.
This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over …
An analytico-geometric reflection principle is established by means of normal deformations of analytic discs.
New distance comparison principle for curve shortening flow in higher dimensions.
problem Understanding curve shortening flow in higher dimensions.
method Established a variant of Huisken's distance comparison principle.
result Symmetric curve shortening flow with one-to-one convex projection develops Type I singularities and becomes asymptotically circular.
Unified framework for Alexandrov 3-spaces, extending manifold results.
problem Extend manifold topology results to Alexandrov 3-spaces.
method Generalize connected sum, prime decomposition, and Dehn surgery.
result Unified framework for Alexandrov 3-spaces, including non-manifold spaces.
Embeds finite Alexandrov spaces into Euclidean products.
problem Embedding Alexandrov spaces into Euclidean products.
method Canonical embedding into a product of Alexandrov and Euclidean spaces.
result Affine functions on X arise from Euclidean functions.
Estimates eigenvalues on differential forms on Alexandrov spaces.
problem Estimating eigenvalues of differential form Laplacians on Alexandrov spaces.
method Using Alexandrov spaces with curvature bounded below, constructing differential form Laplacians, and applying local biLipschitz assumptions.
result The differential form Laplacian has a compact resolvent under local biLipschitz assumption, and its kernel is identified with an intersection homology group.
The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
problem Conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
method Proposes a Gluing Conjecture and proves it under certain conditions.
result The Gluing Conjecture is true under specific conditions, generalizing Petrunin's Gluing Theorem.
We build an elementary analytico-geometric theory of Segre chains and their jets.
Uniform bounds derived for fully non-linear equations.
problem Bounding fully non-linear equations uniformly in background metrics.
method Auxiliary Monge-Ampère equations and entropy-like quantities.
result Uniform L∞ bounds for systems coupling fully non-linear equations to their linearizations. Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
The paper extends an Alexandrov theorem to Minkowski spacetime.
problem Generalizing Alexandrov's theorem to Minkowski spacetime.
method Adapting conditions for closed codimension-two spacelike submanifolds in Minkowski spacetime.
result A generalized Alexandrov theorem for Minkowski spacetime.
We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…
In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Le…
Classifies 4D Alexandrov spaces with torus actions.
problem Classifying Alexandrov spaces with torus actions.
method Equivariant classification and homeomorphism analysis.
result Alexandrov spaces are homeomorphic to Riemannian orbifolds.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
The paper improves collapsing Alexandrov spaces results using good coverings.
problem Collapsing Alexandrov spaces with no proper extremal subsets.
method Application of good coverings of Alexandrov spaces.
result Construction of an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups.
Graph comparison ties to Alexandrov's theorems.
problem Graph comparison conditions on metric spaces.
method Proof of Alexandrov's implications from graph comparisons.
result Complete description of graphs with trivial comparisons.