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. 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.
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.
Study on prescribing scalar curvature on spheres using flow method.
problem Prescribing scalar curvature on n-sphere with sign-changing function.
method Scalar curvature flow approach.
result Flow converges to a metric with prescribed sign-changing function.
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.
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.
Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
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.
Study on Yamabe flow for negative scalar curvature.
problem Prescribed scalar curvature problem on compact manifolds.
method Yamabe flow with conditions on scalar curvature function.
result Long time existence and convergence of the flow.
Conditions for scalar curvature on compact manifolds under conformal deformation.
problem Finding conditions for scalar curvature functions on compact manifolds.
method Analyzing sufficient and necessary conditions for scalar curvature problems within conformal classes.
result Conditions for scalar curvature functions on various compact manifolds.
Proves existence of metrics with prescribed Q-curvature in Riemannian manifolds.
problem Existence of metrics with prescribed Q-curvature under natural assumptions.
method Proves existence of metrics with prescribed Q-curvature in Riemannian manifolds.
result Existence of metrics with prescribed Q-curvature in Riemannian manifolds.
Study reconstructs Morse-Bott functions with specific preimage conditions on 3D manifolds.
problem Reconstructing Morse-Bott functions with prescribed preimages on 3D manifolds.
method Conditions and approach based on previous work by Sharko and others.
result New result on reconstruction of nice smooth functions with specified preimages.
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.
Study proves existence of special hypersurfaces in de Sitter space.
problem Existence of compact spacelike hypersurfaces with prescribed curvature.
method Proves existence using prescribed k-curvature in de Sitter space.
result Compact spacelike hypersurfaces with prescribed k-curvature exist in de Sitter space.
In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).
Existence theorem for prescribed Ricci curvature on homogeneous spaces.
problem Finding metrics with prescribed Ricci curvature on homogeneous spaces.
method Variational properties of curvature functionals on G-invariant metrics. result Existence of solutions to the prescribed Ricci curvature problem.
Global existence and convergence proved for Kazdan-Warner equation with non-negative prescribed function.
problem Existence and convergence of solutions to the Kazdan-Warner equation on a closed Riemann surface.
method Global existence and convergence proved using additional assumptions on the prescribed function and the geometry of the surface.
result Global existence and convergence of solutions proved under specific conditions.
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio.
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 fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.
problem Finding conformal Hermitian metrics with prescribed curvature functions.
method Blow-up argument and partial uniform ellipticity.
result Our assumptions are almost sharp, with some geometric function theory obstructions.
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.
The paper solves a curvature problem on a ball's surface near constant values.
problem Prescribing almost constant curvatures on a manifold with boundary.
method Perturbative approach and ansatz by Han and Li.
result New existence results for conformal metrics when curvatures are near constants.
Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.
problem Solving the prescribed Ricci curvature problem for homogeneous metrics.
method Examining global properties of the scalar curvature functional, focusing on its critical points and maximum.
result Conditions for a global maximum of the scalar curvature functional on a general homogeneous space.
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 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.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.
problem Solving complex Hessian equations with specific singularity types on compact Kähler manifolds.
method Analyzes the total mass of complex Hessian measures and solves equations with prescribed singularities.
result Proves non-decreasing total mass of complex Hessian measures and solves complex Hessian equations.
Study on polygons with fixed edge slopes and their perimeter function.
problem Characterizing and analyzing polygons with prescribed edge slopes.
method Configuration space description and perimeter as a Morse function.
result Characterization and computation of critical points and their Morse indices.
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
Proves existence of hypersurfaces with specific curvature properties.
problem Existence of hypersurfaces with prescribed boundary and Weingarten curvature.
method Solves a fully nonlinear elliptic PDE with a priori estimates.
result Proves existence of hypersurfaces under natural assumptions.
The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.
In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.
Study of Kähler metrics with prescribed singularities on complex spaces.
problem Define and study spaces of Kähler potentials with prescribed singularities.
method Define non-pluripolar products, spaces of finite energy Kähler potentials, and metrics.
result Spaces of finite energy Kähler potentials with prescribed singularities are complete metric spaces.
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.
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…
Constructs real algebraic functions with specified preimages.
problem Reconstructing smooth functions with prescribed preimages.
method Using real algebraic functions and techniques from singularity theory and differential topology.
result Constructs examples of real algebraic functions with specified preimages.
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.
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…
In this paper we find strictly locally convex hypersurfaces in Rn+1 with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for L…
The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.
problem Existence of curves with prescribed angles to torse-forming vector fields in Riemannian manifolds.
method Introducing the notion of a prescribed angle curve and proving its existence for torse-forming vector fields.
result Existence of prescribed angle curves in Riemannian manifolds associated with torse-forming vector fields.