The Minimal Genus Problemmath.GT
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.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,657 papers · 148 categories
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48 results for “minimization problem”
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.
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.
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.
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 …
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 $\BHH$.
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 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.
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.
This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after Lawson-Osserman's paper \cite{l-o} on the Dirichlet problem for minimal graphs of high codi…
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.
In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we estab…
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 …
On Santaló's Problemmath.DG
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.
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
method Equivariant min-max theory for -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- 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 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).
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…
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
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 -norm minimization under second moment constraints, and using and 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…
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring …
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.
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.
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.
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.
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].
This paper embeds surfaces in 3D spheres and balls with minimal area.
problem Embed surfaces with boundary in as minimal surfaces.
method Optimizing Laplace and Steklov eigenvalues with symmetry groups.
result Proves existence of minimal surfaces in with area below .
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.
Discrete approximation solves Björling's minimal surface problem.
problem Constructing minimal surfaces from real-analytic curves with specified normal fields.
method Approximate solution by discrete minimal surfaces and discrete isothermic surfaces.
result Approximation error is proportional to the square of the mesh size.
Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.
problem Finding smooth minimizing hypersurfaces in high dimensions.
method Analyzing the Plateau problem and area minimization in integral homology.
result Smooth minimizing hypersurfaces are generic in 11D, with singularities in higher dimensions.
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the ana…
The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.
problem Optimizing metrics for free boundary minimal surfaces in spherical caps.
method Introducing functionals based on eigenvalues of Steklov-type problems and proving maximizers are induced by immersions.
result Maximizing metrics are induced by free boundary minimal immersions in geodesic balls of a round sphere.
The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.
problem Constructing free boundary minimal surfaces in product spaces of balls.
method Extremal eigenvalue approach involving mixed Steklov-Neumann eigenvalues.
result No absolute maximum exists for the problem in product spaces.
In the present paper, the minimal investment risk for a portfolio optimization problem with imposed budget and investment concentration constraints is considered using replica analysis. Since the minimal investment risk is influenced by the investment concentration constraint (as well as the budget constraint), it is i…
New approach to solving minimal surface system Dirichlet problem on smooth domains.
problem Solving Dirichlet problem for minimal surface system on smooth domains.
method Using mean curvature flow (MCF) with boundary conditions and non-negative Ricci curvature assumption.
result Existence of long-time mean curvature flow and existence result for exterior Dirichlet problem.
In this paper we establish a connection between free boundary minimal surfaces in a ball in and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
New algorithm for online convex minimization over integer lattice.
problem Online decision-making with nonlinear combinatorial objectives.
method Introduces online -convex minimization and proposes efficient algorithms.
result Tight regret bound for full information setting algorithm.
The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
Study on spheres with minimal equators.
problem Classifying metrics on spheres with minimal equators.
method Survey and discussion of related problems.
result Classification of metrics on spheres with minimal equators.