The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
problem Uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
method Analyzes criteria for uniqueness and constructs examples of non-uniqueness.
result Uniqueness of minimal surfaces is equivalent to uniqueness in a smaller class of stable minimal disks.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
Unique solution found for even L^p Minkowski problem at p=p0, but fails for p<p0.
problem Uniqueness of solutions for even L^p Minkowski problem.
method Characterization of p0 through Hilbert operator eigenvalue, analysis of convex bodies.
result Existence and uniqueness of solutions for even L^p Minkowski problem at p=p0, failure for p<p0.
Uniqueness of quasi-roots explored in right-angled Artin groups.
problem Uniqueness of quasi-roots in right-angled Artin groups.
method Introducing quasi-roots and studying their uniqueness.
result Uniqueness of quasi-roots established in right-angled Artin groups.
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Alternative proof for static black hole uniqueness with nonpositive mass.
problem Proving static uniqueness for Kottler spacetimes with nonpositive mass.
method Alternative, more elementary proof of static uniqueness theorem.
result Alternative proof of static uniqueness for Kottler spacetimes with nonpositive mass.
New method proves uniqueness in mean curvature flow.
problem Proving uniqueness in mean curvature flow.
method Arguments from [CM2] to get stronger uniqueness.
result Stronger effective version of uniqueness of blowups.
Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.
problem Proving uniqueness of Kähler Ricci shrinkers on toric orbifolds.
method Extending results from toric manifolds to toric orbifolds.
result Uniqueness of Kähler Ricci shrinkers on toric orbifolds established.
Proves uniqueness of catenoid-like shapes in a ball.
problem Uniqueness of catenoid-like minimal surfaces.
method Analyzes σ-homothetic free boundary minimal annuli. result Critical catenoid is the only σ-homothetic shape. The paper assesses conditions for the uniqueness of k-means clustering.
problem Conditions for the uniqueness of k-means clustering.
method Analyzes the choice of k and provides necessary and sufficient conditions for uniqueness.
result Determines the asymptotic distribution of the within cluster sum of squares (WCSS) and provides a bootstrap test for uniqueness.
Study end sum for surfaces and prove uniqueness results.
problem Uniqueness of end sum for surfaces and related manifolds.
method Analyzing end sum and adding a 1-handle at infinity for surfaces.
result The end sum of two surfaces with compact boundary is uniquely determined by the chosen ends.
We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of Lp solutions with 0<p<1, and improves the L1 uniqueness result of P. Li by weakening the curvature assumption.
Non-unique option pricing in Heston model analyzed mathematically.
problem Non-uniqueness of call option prices in the Heston model.
method Analysis of degenerate parabolic equations in the context of option pricing.
result Construction of a new example demonstrating the accuracy of a uniqueness theorem.
Paper proves unique conical tangent flows in all dimensions.
problem Proving uniqueness of conical singularities in mean curvature flows.
method Based on Chodosh-Schulze's work in hypersurfaces, extended to all codimensions.
result Uniqueness of asymptotically conical tangent flows in all codimensions.
Simple sphere eversion with a unique point.
problem Sphere eversion with a unique quadruple point.
method Inspired by Morse theory, without homotopical tools.
result Unique quadruple point in sphere eversion.
Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. The paper proves uniqueness of a solution in general relativity.
problem Uniqueness of solutions in the conformal method for Einstein's constraint equations.
method Analyzes solutions with arbitrary mean curvature and volume constraint.
result The Holst-Nagy-Tsogtgerel--Maxwell solution is unique for volumes below a certain threshold.
Unique incompressible surface in fibered 3-manifolds.
problem Finding unique incompressible surfaces in fibered 3-manifolds.
method Construction of a family of fibered hyperbolic 3-manifolds.
result The unique incompressible surface is the fibre itself.
Curve shortening flow is not unique on certain metrics.
problem Non-uniqueness of curve shortening flow on specific metrics.
method Formulated a uniqueness conjecture and constructed a non-static solution.
result Curve shortening flow is not unique on a non-flat metric on the plane.
We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map u:C→H2 satisfying ∂u=0 with prescribed polynomial Hopf differential; there is a unique affine spherical imm…
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second the…
Proves uniqueness of certain S1-symmetric gravitational instantons.
problem Proving uniqueness of S1-symmetric gravitational instantons. method Using a divergence identity and results from the G-signature theorem. result Proof of the S1-symmetric Euclidean Black Hole Uniqueness conjecture. Revisits geometric PDE uniqueness in Riemannian and CR geometry.
problem Uniqueness of solutions to geometric PDEs in Riemannian and CR geometry.
method New proofs and reconstruction of Jerison-Lee identity.
result Stronger uniqueness result in CR geometry.
Unique minimal surfaces near quadratic cones are identified.
problem Identifying minimal surfaces near quadratic cones.
method Analyzing minimal hypersurfaces inside the unit ball with perturbed boundary conditions.
result Minimal surfaces are uniquely determined by their boundary conditions.
Unique extremal Kähler metric found near a divisor.
problem Uniqueness of extremal Kähler metric near a smooth divisor.
method Analyzes Poincaré type extremal Kähler metric with cusp singularity.
result Uniqueness of extremal Kähler metric up to holomorphic transformations.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
Unique floating and buoyancy surfaces identify convex polytopes.
problem Identifying convex polytopes from their flotation and buoyancy surfaces.
method Proving uniqueness of surfaces for polytopes with uniform or prescribed density.
result Floating and buoyancy surfaces uniquely determine convex polytopes.
Paper proves uniqueness of a complex construction.
problem Proving uniqueness of a complex construction.
method Defined a total complex and proved uniqueness of terms.
result Total complex CTot(L) is unique. In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Mo…
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
Unique maximal curve systems found for up to 5 punctures.
problem Finding unique maximal curve systems in punctured projective planes.
method Analyzing mapping class group action on maximal 1-systems of loops. result Maximal 1-system is unique for up to 5 punctures. Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
Unique solutions found for diffusive martingale problems.
problem Finding unique solutions to Cauchy problems for diffusive real-valued strict local martingales.
method Provided sets of smooth functions under local Hölder and Engelbert-Schmidt conditions for unique classical and weak solutions.
result Unique solutions found for specific martingale models.
The paper proves uniqueness of Ricci flow on noncompact manifolds.
problem Proving uniqueness of Ricci flow on noncompact manifolds.
method Pseudolocality theorem for L-complete Ricci flow. result Strong uniqueness theorem for L-complete Ricci flow on Euclidean space. We show that zero-Maslov class Lagrangian self-expanders in C^n which are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.
Paper proves uniqueness of Type II Yamabe metrics on manifolds.
problem Uniqueness of Type II Yamabe metrics on compact manifolds.
method Investigates sufficient conditions for metric uniqueness and proves corresponding theorems.
result Establishes sufficient condition for a metric to be the unique Type II Yamabe metric.
The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.
problem Proving uniqueness of entire graphs evolving by mean curvature flow.
method Analyzes graphs of locally Lipschitz functions and rotationally symmetric solutions, proving uniqueness under uniform lower bounds and proper graphs.
result Uniqueness of entire graphs evolving by mean curvature flow under specified conditions.
New research proves uniqueness of maximal spacetime boundaries under certain conditions.
problem The uniqueness of maximal spacetime boundaries in extendible spacetimes.
method Analyzing manifolds with boundary, excluding specific geodesic behaviors, to prove uniqueness.
result Extendible spacetimes admit a unique maximal future boundary extension under suitable assumptions.
New algorithm detects unique events in time series data.
problem Detecting anomalous events in unknown properties.
method Model-free, unsupervised detection using Temporal Outlier Factor (TOF).
result TOF outperforms traditional outlier detection methods.
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
Euler's elastica with monotone curvature is uniquely minimal.
problem Global minimality of planar elastica with monotone curvature.
method Proof of global minimality using clamped boundary conditions and length penalization.
result Every planar elastica with non-constant monotone curvature is uniquely minimal.
The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.
problem Uniqueness of factorization of knotted handlebodies along decomposing 2-spheres.
method Analyzes factorization of knotted handlebodies in the 3-sphere, proving uniqueness for specific cases.
result Determines chirality of 6_{10} handlebody-knot and constructs an infinite family of hyperbolic handlebody-knots.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.