arXiv research

The paper confirms Ilmanen's conjecture about mean curvature flows.

Paper resolves Huisken's conjecture without strict genus drop theorem.

Proves multiplicity one for min-max minimal hypersurfaces in specific manifolds.

Proves multiplicity one for mean curvature flow singularities.

Proves multiplicity one conjecture for surface flows, with applications in flow regularity.

Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.

In Riemannian manifolds, minimal hypersurfaces with large area exist or have complex structures.

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are ab…

Strictly stable Allen-Cahn hypersurfaces have multiplicity one.

Four minimal spheres found in sphere with special metric.

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

Proves a generalization of a multiplicity one theorem for specific groups.

Study shows strong multiplicity one property for 3D hyperbolic spaces.

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

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the exi…

The paper shows mean curvature flow keeps diameter bounded under certain conditions.

Stable solutions to a specific equation are one-dimensional.

We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-…

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this constr…

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible fl…

Proves mean convex neighborhood conjecture for ancient flows near singularities.

Uniqueness of conical flows helps understand singularities in surface flows.

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.

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

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singu…

Polynomial algorithm for multiplication on one-hole torus skein algebra.

We find sharp upper bounds for the multiplicities and the numerical values of all the distinct eigenvalues on a surface of revolution diffeomorphic to the sphere.

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

The study examines mass drop and multiplicity in mean curvature flow.

Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and Whit…

We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chose…

CAOS aggregates multiple one-shot predictors for efficient uncertainty quantification.

We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.

The paper proves a regularity theorem for Brakke flows near triple junctions.

Study shows uniform decay rate for singular mean curvature flows.

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover…

Study shows generic surfaces avoid complex flow patterns.

The paper studies minimal graphs with bounded 2-dilation in Euclidean space.

This paper develops a new nonlocal approximation method for minimal surfaces, proving robust estimates and separation properties.

Minimal surfaces in lens spaces identified with specific counts.