arXiv research

Definite knots' quotients remain definite via Seifert surfaces.

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

Constructs a unique surface in a ball with specific properties.

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

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

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.

We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by Pitts-Rubinstein. These …

The paper defines new knot genera and finds bounds for stabilization distances.

New invariants refine link homology, showing large genus differences.

Study on knots, genera, and algebraic concordance groups.

Study equivariant 4-genus of knots in symmetric 4-manifolds.

Maps on surfaces can be embedded into spheres with minimal dimensions.

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…

Study knot Floer homology to create concordance invariants and slice genus bounds.

For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $Σ_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $Σ_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $Σ_g$ converge as varifolds to the union of the d…

The paper proves an adjunction inequality for Real embedded surfaces in 4-manifolds.

New minimal hypersurfaces in 4D sphere found.

A projection maps geodesic currents to Teichmüller space.

Random spherical harmonics on $S^3$ have a single nodal component with expected genus proportional to $mN$.

New minimal surface theory disproves a conjecture in symmetric spaces.

New spectral estimates for minimal surfaces with boundary conditions.

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construc…

The paper finds a special hypersurface in a manifold with positive Ricci curvature.

The Witten class is derived from equivariant cohomology of a conformal loop space.

Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.

New findings on knot genera using advanced techniques.

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space…

Minimal dimensions for Riemann surface embeddings computed for specific groups.

In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb …

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

Generalizes Molino's theory for Riemannian foliations.

The paper calculates the equivariant genus for a specific type of knot.

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

Optimizes metrics on surfaces for eigenvalues.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such …

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

By using the equivariant localization formula of toric varieties. We prove the vanishing of the Witten genus of some string complete intersections in smooth toric varieties.

Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sit…

New knot concordance invariants from Seiberg-Witten theory bound slice genus.

Study of free boundary minimal surfaces with new existence and degeneration results.

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $ρ: π_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric spa…

If a finite group of orientation-preserving diffeomorphisms of the 3-dimensional torus leaves invariant an oriented, closed, embedded surface of genus g>1 and preserves the orientation of the surface, then its order is bounded from above by 12(g-1). In the present paper we classify (up to conjugation) all such group ac…

Outer space and Teichmüller space fail well-rounded retract analogy.

Minimal surfaces in hyperbolic 4-space degenerate to core of product trees.

Bound on equivariant index for min-max surfaces.

New minimal discs and annuli found in ellipsoids.