arXiv research

Max flow/min cut theorem extended to currents and topology.

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow…

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for net…

A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This gives a relative version of Stallings' Theorem on the structure of groups with mo…

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, …

Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.

In this paper it is shown that for any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair $s, t$ where each of $s, t$ is either a vertex or an end in the original network. A Max-Flow Min-Cut Theorem is proved for any net…

New self-expander found between two given asymptotic ones.

The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.

New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.

Localized min-max method proves minimal hypersurface existence.

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.

The study proves a generic multiplicity one theorem for $G$-invariant minimal hypersurfaces.

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…

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the …

Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.

Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.

We consider the problem of learning optimal binary classification trees. Literature on the topic has burgeoned in recent years, motivated both by the empirical suboptimality of heuristic approaches and the tremendous improvements in mixed-integer programming (MIP) technology. Yet, existing approaches from the literatur…

5 minimal tori found in 3-spheres with positive Ricci curvature.

Paper defines a new functional for spinors on Euclidean manifolds.

The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.

Theory for capillary surfaces in 3-manifolds with smooth boundary.

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this …

Study shows instability of specific cone solutions in high-dimensional spaces.

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper "the asymptotics of ECH capacities" . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max…

New theorem finds new minimal hypersurfaces in hyperbolic space.

Let $(M,g)$ be a compact Ricci-flat 4-manifold. For $p \in M$ let $K_{max}(p)$ (respectively $K_{min}(p)$) denote the maximum (respectively the minimum) of sectional curvatures at $p$. We prove that if $K_{max} (p) \le \ -c K_{min}(p)$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac{2+\sqrt 6}{4}$, th…

The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.

Existence proof of noncompact self-shrinkers with arbitrary genus.

Study introduces statistical mechanics for min-max problems.

Study shows strong min-max principle for phase transitions.

Equity-Transformer solves NP-hard min-max routing problems efficiently.

Upper bound for Morse index of min-max varifolds.

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture …

The paper characterizes eigenvalues of surfaces using min-max quantities for harmonic maps.

The study proves the existence of free boundary minimal disks in convex regions.

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated…

Adaptive momentum method solves non-convex min-max problems.

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

Robust VB framework handles contamination using min-max median aggregation.

Paper improves Morse index bound for hypersurfaces.

Infinite $G$-invariant minimal hypersurfaces found in Riemannian manifolds.

Study on local convergence of min-max algorithms to differential equilibria on Riemannian manifolds.