Study area minimizing currents in conformal cones, solving Dirichlet problems.
problem Area minimizing currents in conformal cones.
method Minimizing problem of area functionals, Dirichlet problem of minimal surface equations.
result Existence and uniqueness of area minimizing currents in a wide class of conformal manifolds.
Survey of minimal genus problem progress.
problem Finding the smallest genus surface in 3-manifolds.
method Survey and review of existing research.
result Summarizes recent advancements in the minimal genus problem.
Solves risk minimization problem with SSD constraints.
problem Finding SSD-minimal quantile function under mixed constraints.
method Explicitly works out SSD-minimal solution and relates to Skorokhod problem.
result Explicit solution to risk minimizing problem.
Extends Newton's minimal resistance problem to Riemannian surfaces.
problem Minimal resistance on Riemannian surfaces.
method Derive resistance functional, analyze constrained minimization.
result Smooth extremals are loxodromes, global minimizers characterized.
Solves minimal surface problem in H^2xR.
problem Minimal surfaces in H^2xR with specified asymptotic boundary.
method Identifies finite Jordan curves bounding minimal surfaces.
result Identifies specific Jordan curves for minimal surfaces.
Recent progress on minimal surface system and cones in Euclidean spaces.
problem Exploring the Dirichlet problem for minimal surfaces and cones.
method Systematic developments and new families of minimizing cones.
result New families of minimizing cones of different types.
The paper solves area minimizing problems in special geometric cones.
problem Area minimizing problems in conformal cones.
method Defining NCM condition, proving existence of minimal graphs, solving in specific cones.
result Existence of minimal graphs in mean convex conformal cones.
Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
The paper connects minimal surfaces to cones in a ball.
problem Understanding minimal surfaces with free boundaries.
method Establishing a connection between minimal surfaces and cones.
result Doubly connected minimal surfaces in a ball are catenoids.
Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a …
Paper shows non-existence of solutions for minimal surface problems.
problem Non-existence of solutions to Dirichlet problems for minimal surfaces.
method Analyzes minimal graphs of codimension ≥2 over domains with possibly non- C 1 C^1 C 1 boundaries. result Proves non-existence of solutions in various scenarios.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$ .
result Fairly complete solution for finite curves in $\BHH$ .
New convex programs solve minimal-area problems on Riemann surfaces.
problem Finding the conformal metric of least area with constraints on curve lengths.
method Formulated as a local convex program using calibrations and max flow-min cut theorem.
result Derives new insights and numerical solutions for minimal-area metrics.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for λ λ λ above which minimizers touch the obstacle, regardless of obstacle shape. Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
New eigenvalue problem for free boundary minimal surfaces in a ball.
problem Eigenvalue problem for free boundary minimal surfaces in a unit ball.
method New eigenvalue problem associated with the Jacobi operator.
result Eigenfunctions of the unit normal are associated with eigenvalue -2.
Research on the least complex surface in certain 4D shapes.
problem Finding the simplest surface in specific four-dimensional shapes.
method Analyzing smooth four-manifolds to determine minimal genus.
result Results on the minimal genus for studied four-manifolds.
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.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
Paper tackles Santaló's convex surface problem in hyperbolic 3-space.
problem Characterize convex surfaces minimizing total mean curvature with fixed area.
method Proposes conjectural minimizer description and constructs new surface candidates.
result Establishes property of singular points of any minimizer.
Proves existence and uniqueness of solutions for a specific minimal surface equation.
problem Existence and uniqueness of solutions for a singular minimal surface equation.
method Proves existence and uniqueness of classical solutions for the Dirichlet problem.
result Proves existence and uniqueness of solutions for the α α α -singular minimal surface equation. Study determines minimal surfaces from boundary data, proving topological and conformal recoverability.
problem Determining minimal surfaces from boundary data.
method Developed a semiclassical nonlinear calculus for complex geometric optics solutions.
result Minimal surfaces can be recovered from the Dirichlet-to-Neumann map under certain conditions.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in S 4 \mathbb{S}^4 S 4 method Equivariant min-max theory for G G G -invariant minimal hypersurfaces result Construction of embedded non-equatorial minimal hypersphere
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
In this paper we study general Schatten- p p p quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized v e c t o r vector v ec t or minimization problem are equiva…
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Solves a Cauchy problem for minimal spacelike surfaces in 4D spacetime.
problem Constructing minimal spacelike surfaces in 4D spacetime.
method Defining isoclinic parametric surfaces and proving their relation to holomorphic functions, solving the Cauchy problem.
result Solves the Cauchy problem for minimal spacelike surfaces in R 2 4 \mathbb{R}^4_2 R 2 4 . We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau prob…
Minimal surfaces in Heisenberg group solved via Dirichlet problems.
problem Existence of minimal surfaces in Heisenberg group with balanced metric.
method Solving Dirichlet problems for minimal surface equation.
result Existence of complete properly embedded minimal surfaces.
Study on minimal surfaces in a specific homogeneous space with non-existence and construction results.
problem Minimal surfaces in S L ~ 2 ( R ) {\widetilde{\mathrm{SL}}_2(\mathbb{R})} SL 2 ( R ) with asymptotic boundary conditions. method Non-existence proofs and construction of specific minimal surfaces.
result Existence and non-existence results for minimal surfaces in S L ~ 2 ( R ) {\widetilde{\mathrm{SL}}_2(\mathbb{R})} SL 2 ( R ) . Characterizes kernel of linearization for minimal surfaces problem
problem Characterizing kernel of linearization for minimal surfaces problem
method Show kernel consists of potential fields and TT fields
result In whole-space Euclidean decomposition, kernel consists of potential fields and TT fields
Paper proposes a new method to minimize submodular functions with fewer calls to simpler oracles.
problem Minimizing the sum of submodular set functions with limited information.
method Introduces a modified convex problem requiring constrained total variation oracles that can be solved with fewer calls to minimization oracles.
result Shows significant reduction in the number of calls to minimization oracles.
New algorithms improve submodular minimization via DC programming.
problem Minimizing the difference of two submodular functions.
method Introducing variants of the DC algorithm (DCA) and its complete form (CDCA) for DC programs corresponding to DS minimization.
result Our algorithms outperform existing baselines on speech corpus selection and feature selection.
New method estimates robust mean in high dimensions with minimized outliers.
problem Estimating the mean in high dimensions when a fraction of data is corrupted.
method Formulating the problem as ℓ 0 \ell_0 ℓ 0 -norm minimization under second moment constraints, and using ℓ 1 \ell_1 ℓ 1 and ℓ p \ell_p ℓ p minimization techniques. result The proposed method achieves order optimal robust mean estimation and significantly outperforms existing methods.
We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in H2xR. In particular, we identify the collection of Jordan curves in the asymptotic boundary of H2xR, which bounds an area minimizing surface in H2xR. Furthermore, we study the similar problem for minimal surfaces, and s…
Paper connects minimal and maximal surfaces' boundary value problems.
problem Existence and uniqueness of minimal surfaces' boundary value problems.
method One-to-one correspondence between minimal and maximal surfaces' solutions.
result One-to-one correspondence between minimal and maximal surfaces' boundary value problems.
Study minimizes risk in MDPs with spectral measures.
problem Minimizing risk in MDPs with spectral measures.
method Splitting into inner and outer minimization problems; solving inner as MDP; proving existence for outer.
result Existence and solution methods for the outer minimization problem.
New geometric insights reveal properties of adversarial training problems.
problem Adversarial training in binary classification.
method Equivalence with regularized risk minimization and convex relaxations.
result Existence of minimal and maximal solutions, and regular solutions.
Minimal networks minimize length and mass in certain configurations.
problem Finding minimal networks that minimize length and mass.
method Global and local calibrations to prove minimization properties.
result Minimal networks minimize mass and interfaces in partitions.
Study determines a minimal surface in a Riemannian manifold from boundary data.
problem Determining a minimal surface in a Riemannian manifold from boundary data.
method Analyzes the Dirichlet-to-Neumann map for the minimal surface equation.
result Knowledge of the Dirichlet-to-Neumann map determines the Riemannian manifold up to isometry.
Minimal genus surfaces solve homology problems in finite complexes.
problem Finding surface representatives of homology classes with minimal genus.
method Minimizing genus and Euler characteristic, analyzing surgeries and homotopy.
result Minimizers are homotopic to cellwise coverings, but problem is undecidable in general.
Extends Newton's minimal resistance problem to Lorentz-Minkowski space.
problem Minimal resistance in Lorentz-Minkowski space.
method Derived functional energy, determined Euler-Lagrange equation, analyzed maximum principle, found separable and radial solutions.
result Obtained solutions with conical singularities at the origin and analyzed the Single Shock Condition.
Study proves conditions for area-minimizing surfaces in a specific space.
problem Existence and non-existence of area-minimizing surfaces in E ( − 1 , τ ) \mathbb{E}(-1,τ) E ( − 1 , τ ) . method Analyzes sufficient conditions for curves to be the asymptotic boundary of area-minimizing surfaces.
result Presented sufficient conditions for a curve to admit a solution to the asymptotic Plateau problem.
Hexagonal norm double bubble problem solved with minimal configurations.
problem Finding the optimal shapes for minimizing perimeter in hexagonal geometry.
method Elementary proof and geometric exclusions to simplify minimizer search.
result Existence of minimizing sets for volume ratio parameter α in (0,1].
New limits of minimal surface systems have surprising large interior parts.
problem Minimal surface system limits with large interior vertical and non-minimal portions.
method Construction of limits with smallest possible dimension and codimension.
result Limits of minimal surface systems can have surprising large interior parts.
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.
A meta-learning approach improves the performance of alternating minimization for non-convex optimization problems.
problem Optimizing non-convex problems with multiple variables using alternating minimization.
method Meta-learning based alternating minimization (MLAM) to replace handcrafted updating rules.
result The proposed MLAM method outperforms traditional AM-based methods in various non-convex optimization problems.