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 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. 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.
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…
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…
We use a phase space analysis to give some classification results for rotational hypersurfaces in Rn+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in Sn, we show that a Delaunay-type classification hold…
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.
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.
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.
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.
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.
We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.
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.
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.
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.
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.
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.
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…
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.
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.
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 …
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 prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.
We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.
We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to Sn for small n,n≤6, provided there are barriers.
The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.
problem Existence of H-spheres with arbitrary codimensions in closed Riemannian manifolds.
method Min-max theory and Morse index analysis.
result Existence of branched immersed H-spheres with controlled Morse index and arbitrary codimensions.
Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.
Paper finds unique solutions for curved surfaces with specific gradient.
problem Existence of curved surfaces with specific gradient.
method Second boundary value problem of constant mean curvature equations.
result Unique convex solutions for constant mean curvature equations.
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
Study translating solitons in Minkowski space with prescribed Gauss image.
problem Second boundary value problem for mean curvature flow in Minkowski space.
method Construct translating solitons with prescribed Gauss image.
result Constructing translating solitons with prescribed Gauss image in Minkowski space.
We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…
Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.
Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
Study minimizes Willmore energy with constraints on surface properties.
problem Minimizing Willmore energy under specific surface properties.
method Adapting Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle methods.
result Existence of smooth minimizers for a broad range of constraints.
New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case n≥5 under pinching conditions. Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.