arXiv research

The paper bounds the min-max width of embedded circles on spheres and manifolds.

We accelerate min-max optimization and apply it to minimal bounding sphere problems.

Proves existence of special 2-spheres in curved 3-spaces.

We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…

3D spheres can't be swept by short curves, complicating geodesic length estimates.

Study bounds the Morse index of a special torus to 1.

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

New minimal surfaces found in 3D sphere with low genus.

New minimal hypersphere found in 4-sphere solving Bernstein problem.

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

New geometric invariant from min-max width of spheres on Riemannian 2-spheres.

Study confirms a 2-sphere metric with three geodesics of minimal length.

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

Improved eigenvalue bounds for minimal hypersurfaces in spheres.

Proves existence of at least two minimal spheres in any 3D space.

Proves existence of curves with constant curvature in a sphere.

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

Four minimal spheres found in sphere with special metric.

Proves conjecture about sphere widths under rotational symmetry.

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

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 …

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4π$ and index at most one…

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Sm…

The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.

The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coin…

Upper bound for Morse index of min-max varifolds.

Bound on equivariant index for min-max surfaces.

Paper improves Morse index bound for hypersurfaces.

New minimal hypersurfaces in 4D sphere found.

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

New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.

The study finds an infinite number of minimal surfaces in 3D spheres.

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 …

We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.

We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12π$, only three of them …

For a Riemannian metric $g$ on the two-sphere, let $\ell_{\min}(g)$ be the length of the shortest closed geodesic and $\ell_{\max}(g)$ be the length of the longest simple closed geodesic. We prove that if the curvature of $g$ is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g…

This paper bounds min-entropy leakage for Blowfish privacy using graph symmetries.

Lower bounds found for nonconvex-strongly-concave min-max optimization problems.

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

This research proves that two min-max theories for hypersurfaces are equivalent.

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N)\ge min (σ(M),Λ) for Λ>0. We derive explicit lower bounds for Λin dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),…

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

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

New bandit problem for finding best group of arms with worst mean reward.

We compute the $k$-width of a round $2$-sphere for $k=1,\ldots,8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold.