Solves second boundary value problem for affine mean curvature equation and related Monge-Ampère equations.
problem Second boundary value problem of prescribed affine mean curvature equation and related Monge-Ampère equations.
method Boundary regularity theory of Monge-Ampère and linearized Monge-Ampère equations.
result Solves the second boundary value problem for the prescribed affine mean curvature equation.
The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-We…
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.
The paper classifies and finds surfaces with specific curvature in isotropic spaces.
problem Finding surfaces with prescribed Gaussian and mean curvature in isotropic spaces.
method Classified and found surfaces of different types with specific curvature conditions.
result Affine factorable surfaces of different types with prescribed curvature were found and classified.
Theory proves existence of hypersurfaces with prescribed curvature.
problem Existence of hypersurfaces with prescribed mean curvature in noncompact manifolds.
method Developed min-max theory for noncompact manifolds.
result Proved existence of closed and finite area hypersurfaces.
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
Study classifies rotational hypersurfaces with prescribed mean curvature.
problem Classifying rotational hypersurfaces with prescribed mean curvature.
method Phase space analysis to classify hypersurfaces.
result Delacunay-type classification for even prescribed functions.
The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
Proves existence of graph on torus with prescribed curvature.
problem Existence of graphs with prescribed mean curvature on torus.
method Proves existence using graph theory and curvature conditions.
result Existence of a graph on the n-dimensional torus with prescribed curvature.
Study on t-graphs with prescribed mean curvature in Heisenberg groups.
problem Existence and uniqueness of t-graphs with prescribed mean curvature. method Characterization of classical solutions without Dirichlet boundary data, conditions for uniqueness, approximation technique for non-constant mean curvature.
result Conditions for existence and uniqueness of t-graphs in Heisenberg groups. Equations link metrics with tensors, revealing curvature constraints.
problem Understanding curvature properties of geometric structures.
method Formal analogies to Einstein-Maxwell equations, studying Codazzi and conformal Killing equations.
result Constraints on scalar curvature of metrics in solutions.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…
Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.
problem Short time existence and long time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
method Finding sufficient conditions for short time existence and discussing long time existence and convergence.
result Sufficient conditions for short time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
Affine spheres can be realized with a specific Blaschke metric condition.
problem Realizing affine spheres with a prescribed Blaschke metric.
method Proving the necessary and sufficient condition for local realizability.
result The equality Δln∣κ−λ∣=6κ is a condition for local realizability of the metric as an affine sphere's Blaschke metric. Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
Develops global theory for surfaces with prescribed mean curvature.
problem Finding complete hypersurfaces with prescribed mean curvature.
method Global theory extending constant mean curvature and self-translating solitons.
result A priori height and curvature estimates for n=2. Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
problem Surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
method Classification result for rotational surfaces with prescribed mean curvature.
result Existence of embedded tori as counterexamples to the Alexandrov problem.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…
Study classifies non-removable singularities in Minkowski 3-space equations.
problem Classifying non-removable singularities in Minkowski 3-space equations.
method Classification through real analytic solutions of the prescribed mean curvature equation.
result Classification of non-removable isolated singularities.
Lectures on PDEs for creating surfaces with specific curvature.
problem Creating surfaces with a specific mean curvature.
method Solving a quasilinear elliptic PDE to find surfaces.
result Graphs with specific mean curvature can be found using PDEs.
Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
Study proves existence of curvature-preserving metrics on balls with symmetries.
problem Existence of conformal metrics with prescribed mean curvature on balls.
method Flow method to prove existence of solutions.
result Existence of conformal metrics with prescribed mean curvature on unit balls with symmetries.
Compact capillary hypersurfaces with specific curvatures are proven.
problem Proving compactness of capillary hypersurfaces with prescribed mean curvature.
method Extending curvature varifolds to oriented integral varifolds and applying curvature varifolds with capillary boundary.
result Compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions.
Survey on rigidity results for graphs with prescribed mean curvature.
problem Rigidity of graphs with prescribed mean curvature.
method Analysis of mean curvature operator, maximum principles, gradient estimates.
result Detailed geometric applications, including Bernstein theorem and splitting theorem.
Generic min-max theory proves existence of hypersurfaces with specific mean curvature.
problem Proving existence of hypersurfaces with prescribed mean curvature for generic functions.
method Generic min-max theory applied to smooth prescription functions.
result Existence of nontrivial, smooth, closed hypersurfaces with specific mean curvature.
Mean curvature flow converges to a translating soliton with prescribed contact angle.
problem Mean curvature flow with contact angle constraints in non-Euclidean settings.
method Existence proof using translating solitons and bounds on convexity and Ricci curvature.
result Graphical solutions converge to a translating soliton as time goes to infinity.
Paper studies surfaces in 3D space with prescribed mean curvature.
problem Understanding surfaces with prescribed mean curvature in R3. method Analyzes behavior of 1-parameter family of annuli at infinity.
result Obtains half-space theorems for surfaces with prescribed mean curvature.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.
The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
problem Finding closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
method One-parameter prescribed mean curvature min-max theory.
result Closed hypersurfaces with prescribed mean curvature are found in certain non-compact manifolds.
Let M be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function H (with mild conditions) on M, we construct a closed incompressible surface with mean curvature H . A direct application is the existe…
Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.
problem Existence and nonexistence of annular surfaces with prescribed mean curvature.
method Proves existence and nonexistence results for normal graphs of unduloids or nodoids.
result Existence and nonexistence results for annular type surfaces with prescribed mean curvature.
It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.
Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.
problem Finding sphere foliations with prescribed mean curvature on Riemannian manifolds.
method Proves existence of foliations by spheres with mean curvature proportional to a given function on non-degenerate critical points.
result Essentially unique foliation of spheres with prescribed mean curvature exists in a neighborhood of a non-degenerate critical point.
Explicit formula found for certain curved surfaces.
problem Finding surfaces with specific curvature properties.
method Provided an explicit formula for singular surfaces of revolution.
result Conditions for singularities of surfaces are given.
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.
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.
Solutions found for specific curvature problems in geometric settings.
problem Finding hypersurfaces with prescribed mean curvature in geometric settings.
method Proving existence of solutions to the asymptotic Plateau problem in Cartan-Hadamard manifolds.
result Existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature.
The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is t…
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.
Study on prescribing mean curvature for S2 to R3 using spinor fields.
problem Prescribing mean curvature for S2 to R3. method Variational methods and spinorial Weierstraß representation.
result Solutions to the spinor field equation on S2. Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.
problem Proving compactness for hypersurfaces with prescribed mean curvature.
method Using oriented integral varifolds and a weak notion of curvature coefficients.
result Locally uniform bounds on second fundamental form lead to compactness.
We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …
It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.