arXiv research

Bound on equivariant index for min-max surfaces.

New minimal surfaces found in ball with boundary constraints.

In this paper we continue our study of equivariant minimal Lagrangian surfaces in $\mathbb{C} P^2$, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

In this note, we present a new look at translationally equivariant minimal Lagrangian surfaces in the complex projective plane via the loop group method.

New minimal discs and annuli found in ellipsoids.

Unstable minimal surfaces in n-space link to hyperbolic products.

Proves conditions for minimal surfaces in complex hyperbolic space.

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

The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.

The study finds many Möbius bands and annuli on toroids.

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

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

Constructs a unique surface in a ball with specific properties.

New spectral estimates for minimal surfaces with boundary conditions.

Minimal dimensions for Riemann surface embeddings computed for specific groups.

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 …

Minimal surfaces with symmetries exist under certain group actions.

New minimal surface theory disproves a conjecture in symmetric spaces.

We construct and analyze minimal disc stackings with bounds on their Morse index.

The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.

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

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

We study symmetric minimal surfaces in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal surfaces in $\mathrm{Nil}_3$ with non-trivial topology. Moreover, we will classif…

New minimal hypersurfaces in 4D sphere found.

Unified rigidity theorem for cyclic and alternating surfaces.

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivari…

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

We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in $\mathbb R^3$. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…

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…

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

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

New Zoll families of minimal spheres found in spheres and projective spaces.

A knot $K$ is definite if $|σ(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces.

A spine is constructed for a non-orientable surface's decorated Teichmüller space.

Minimal surfaces in spheres constructed from symmetry reductions of ODEs.

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

Let $S$ be a closed surface of genus $g \geq 2$ and let $ρ$ be a maximal $\mathrm{PSL}(2, \mathbb{R}) \times \mathrm{PSL}(2, \mathbb{R})$ surface group representation. By a result of Schoen, there is a unique $ρ$-equivariant minimal surface $\widetildeΣ$ in $\mathbb{H}^{2} \times \mathbb{H}^{2}$. We study the induced m…

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…

A projection maps geodesic currents to Teichmüller space.

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…

Given a reductive representation $ρ: π_1(S)\rightarrow G$, there exists a $ρ$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $Σ$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the…

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…

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…

Study of minimal surfaces in a specific symmetric space with polynomial growth.

In this paper a study of $G$-minimality, i.e., minimality of four-manifolds equipped with an action of a finite group $G$, is initiated. We focus on cyclic actions on $CP^2\# \overline{CP^2}$, and our work shows that even in this simple setting, the comparison of $G$-minimality in the various categories, i.e., locally …

The paper bounds the Morse index and nullity of bipolar surfaces related to Otsuki tori.

Study on embedding surfaces into 3-manifolds, focusing on equivariant cases.