Schedule for: 23w6004 - Minimal Surfaces in Symmetric Spaces

Labourie conjectured that given a Hitchin representation of a surface group in a split real Lie group, there exists a unique equivariant minimal surface in the corresponding symmetric space. In my recent work, I show that the analogous conjecture for Fuchsian (Hitchin) representations of a surface group into the semi simple Lie group PSL(2,R)xPSL(2,R)xPSL(2,R) does not hold. Thius implies the existence of non-unique equivariant minimal surfaces inside every Hermitian symmetric space of rank at least three. I also prove that there is no Riemannian metric on the Teichmüller space for which the energy functional is geodesically convex.

Mean curvature flow self-translating solitons are minimal hypersurfaces for a certain incomplete conformal background metric, and are among the possible singularity models for the flow. In the collapsed case, they are confined to slabs in space. The simplest non-trivial such example, the grim reaper curve $\Gamma$ in $\mathbb{R}^2$, has been known since 1956, as an explicit ODE-solution, which also easily gave its uniqueness. We consider here the case of surfaces, where the rigidity result for $\Gamma\times\mathbb{R}$ that we'll show is: The grim reaper cylinder is the unique (up to rigid motions) finite entropy unit speed self-translating surface which has width equal to $\pi$ and is bounded from below. (Joint with D. Impera and M. Rimoldi.) Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (joint with E. Gama & F. Martín), using Morse theory and nodal set techniques, which extend Chini's classification.

For a Higgs bundle over a compact Riemann surface of genus at least 2, the Hitchin-Kobayashi correspondence says the existence of a harmonic metric is equivalent to the polystability of the Higgs bundle. In this talk, we discuss some recent progress on the existence and uniqueness of harmonic metrics on Higgs bundles in the Hitchin section over general non-compact hyperbolic Riemann surfaces. This is joint work with Takuro Mochizuki.

For any metric on a compact surface, the normalized first eigenvalue is bounded above in terms of the topology. This functional is related with several geometric problems and its critical points are given by minimal surfaces of the sphere by the first eigenfunctions. For example, on the torus, the only critical metrics for the first eigenvalue are the Clifford torus on the sphere S^3(1) and the minimal immersion of the flat equilateral torus in the sphere S^5(1), the latter attains the maximum. For genus greater than 1 some examples are known. Maximizing metrics and certain highly symmetric minimal surfaces provide valuable examples of minimal immersion by the first eigenfunctions but the problem is still far from being understood. In particular, it is not clear whether special types of minimal spherical surfaces (say surperminimal or some other condition) could be related to this family. To find sharp estimates for the first eigenvalue functional is another important question. We will present these problems and some recent results on this topic, including new conformal maps of closed Riemann surfaces to the sphere $S^2$ by the first eigenfunctions and explicit estimates for the first eigenvalue in terms of the genus.

Lawson has constructed highly symmetric minimal surfaces of arbitrary genus $g$ in the 3-sphere ${\mathbb S}^3$. I will explain how to construct these surfaces by an integrable system method -- the DPW method. As a byproduct of the construction, we can write their area for large genus $g$ as a series in $1/g$ whose coefficients can be computed in term of values of the Riemann $\zeta$ function and multi-zetas. I will also explain how to estimate the convergence radius of this series. Joint work with L. Heller, S. Heller and S. Charlton.

An immersion $f \colon \Sigma\to \mathbb{CP}^2$ is called a minimal Lagrangian surface if it is minimal with respect to the Fubini study metric and Lagrangian with respect to the K{\"a}hler form. Besides the real projective plane and minimal Lagrangian tori, which can all be constructed via integrable systems methods, the only known compact examples have been obtained by Haskins and Kapouleas for odd genera. In this talk, we explain the construction of new compact minimal Lagrangian surfaces of genus $g=\tfrac{(k-2)(k-1)}{2}$ for large $k\in\mathbb{N}$ using gauge theoretic and loop group factorization methods. These surfaces are analogous to Lawson's minimal surfaces in the 3-sphere and coincide with the projective plane and the Clifford torus for $k=2,3,$ respectively. We determine their symmetry groups and show that the underlying Riemann surfaces are the Fermat curves. We also discuss further geometric properties such as their area and Willmore energy. This talk is based on joint work with Charles Ouyang and Franz Pedit.

I will report on a series of papers, initiated by S. He and myself and continued in the thesis of Dimakis, where large families of solutions of a certain monopole equation on the product of a Riemann surface and a half-line are produced; these families are shown to parametrize some interesting minimal Lagrangian submanifolds in the Hitchin moduli space.

We shall discuss (1) the existence of a harmonic metric for a generic Higgs bundle with a non-degenerate symmetric pairing, (2) the convergence of large-scale solutions of the Hitchin equation, and (3) the estimate of the difference of the Hitchin metric and the semi-flat metric on a Zariski open subset of the moduli space of Higgs bundles. We would like to explain the key roles played by globally or locally defined non -degenerate symmetric pairings in the study of these issues. This talk is partially based on joint works with Qiongling Li and Szilard Szabo.

Higgs bundles played a crucial role in the formulation of Labourie's conjecture and in its proof for rank 2 groups. The key feature in rank 2 is that the minimal surface always lifts to a particular homogeneous space which fibers over the symmetric space (the cyclic space). In this talk I will discuss a class of minimal surfaces in the symmetric space of the split real form of $G_2 $which lift J-holomorphic curves in the pseudo-Riemannian 6-sphere. These surfaces determine (and are determined by) Higgs bundles with a cyclic group symmetry. For $G_2$ Hitchin representations these are exactly the cyclic surfaces considered by Labourie, but in general little is known about the representations which give rise to such surfaces.

I will describe some very old work on stable harmonic maps of surfaces into symmetric spaces. I exhume this material in part because it is pretty, in part because there have been recent developments and finally because there are still some unanswered questions.

Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. Due to the lack of a monotonicity formula, the regularity theory for its critical points, anisotropic minimal surfaces, is much more challenging than the area functional case. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface for elliptic integrands in closed 3-dimensional Riemannian manifolds via min-max construction. This confirms a conjecture by Allard [Invent. Math.,1983] in dimension 3. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.

The 6-sphere has a well-known $G_2$-invariant almost complex structure and its pseudoholomorphic curves have interesting properties: They are minimal (though not homologically minimizing) and the cone on such a curve is an associative 3-fold whose local singular structure is well-understood. I will briefly survey some of the basic known results on the geometry of such pseudoholomorphic curves, in particular exploring their similarities and differences with holomorphic curves in $CP^3$. Using a connection with the $G_2$-invariant holomorphic 2-plane field on the complex 5-quadric discovered by Cartan, I will explain how one can construct pseudoholomorphic curves with some prescribed singularities. If time permits, I will discuss some of what I know about the Gromov compactification of certain moduli spaces of pseudoholomorphic curves in the 6-sphere.

We consider the class of surfaces in a given container with orthogonal constraint on the boundary. We discuss the problem to estimate the area in terms of curvature energy in that class. As an application, we prove the existence of a surface which minimizes the curvature energy.

I will present some recent results regarding hyperbolic three-manifolds with very few number of minimal surfaces.

Let $M$ be a closed hyperbolic manifold containing a totally geodesic hypersurface $S$, and let $N$ be a closed Riemannian manifold homotopy equivalent to $M$ with sectional curvature bounded above by $-1$. Then Besson-Courtois-Gallot proved that $\pi_1(S)$ can be represented by a hypersurface $S'$ in $N$ with volume less than or equal to that of $S$. We study the equality case: if $\pi_1(S)$ cannot be represented by a hypersurface $S'$ in $N$ with volume strictly smaller than that of $S$, then must $N$ be isometric to $M$? We show that many such $S$ are rigid in the sense that the answer to this question is positive. On the other hand, we describe examples of $S,M$ for which the answer is negative.

Minkowski space is endowed with a bordification called the Penrose boundary. In the talk will first give a characterisation of the subsets of Penrose boundary that arise as the set of accumulation points of a properly embedded convex spacelike surfaces in Minkowski space. A primary focus will be directed towards the asymptotic Plateau problem for constant mean curvature (CMC) surfaces and surfaces with constant Gaussian curvature in this particular framework. Through the classical correspondence between CMC surfaces and surfaces with constant Gaussian curvature, these problems are shown to be mutually equivalent. In the second part of the talk, I will delve into the intrinsic completeness of these surfaces. While properly embedded spacelike CMC surfaces in Minkowski space are known to be metrically complete, surfaces of constant Gaussian curvature can exhibit incompleteness. Notably, I will show that these incomplete surfaces are always isometric to domains within the hyperbolic plane, with (maybe empty) geodesic boundaries. Additionally, I will present few results establishing connections between the regularity of the boundary of a surface with constant Gaussian curvature and its intrinsic completeness. This is a joint work with Andrea Seppi and Peter Smillie.

I will review recent and ongoing work on doubling constructions and related results for minimal surfaces. In particular I will present the results in the article ``Generalizing the Linearized Doubling approach , I: General theory and new minimal surfaces and self-shrinkers’’,Camb. J. Math. (to appear); arXiv:2001.04240v4’', by Kapouleas-McGrath, including a new general area estimate for doublings. I will also propose some open questions on which I am working. Finally I will recall results in ``The index and nullity of the Lawson surfaces $\xi_{g,1}$’’, Camb. J. Math. 8 (2020), 363-405, by Kapouleas-Wiygul, and compare to ongoing work by Kapouleas-Zou on the index and nullity of minimal surface doublings of the equatorial two-sphere in the round three -sphere.

I will talk about joint work with Jaoquin Perez at the University of Granada. For all non-negative integers $I$, numbers $H, c >0$ and $K$ non-negative, consider the space $L(I,H,c,K)$ of all complete immersed surfaces $F$ of non-negative constant mean curvature $H$ at most $H$ and Morse index $I$ at most $I$ in complete Riemannian 3-manifolds $X$ with injectivity radius at least $c$ and absolute sectional curvature at most $K$. The Hierarchy Structure Theorem describes how the immersed $H$-surface $F$ organizes itself around any point p in $F$ whose norm of its second fundamental form is sufficiently large and almost maximal near $p$. We shall discover that there is an explicit hierarchical local structure around every such point $p$ that allows one to estimate the index $I$ of $F$ in terms of invariants of local geometry and topology of this hierarchy structure around the collection of all such special points p. In a natural sense, the Hierarchy Structure Theorem allows one to fully understand the local geometry of $F$ around $p$ and to give a complete computational generalization of the Chodosh-Maximo and Karpukhin geometric estimates from below for the index of a complete, finite total curvature, finitely-branched minimal surface $F$ in three-dimensional Euclidean space.

I will explain joint work with N. Sagman comparing the index of large-area minimal surfaces in symmetric space of non-compact type with the index of a related surface in Euclidean space, or equivalently in an affine building. As a consequence, we disprove Labourie's original uniqueness conjecture.

I will talk about the structure and parametrization of the space of free boundary minimal annuli in the Euclidean unit ball. This description is similar to the one given by Pinkall-Sterling and the DPW method in the case of Minimal tori in the 3-sphere. This picture give us new immersed examples (a recent result with Mira-Fernandez) and explain symmetry and compactness properties.

The Asymptotic Plateau Problem is the problem of existence of submanifolds of vanishing mean curvature with prescribed boundary “at infinity”. It has been studied in the hyperbolic space, in the Anti-de Sitter space, and in several other contexts. In this talk, I will present the solution of the APP for complete spacelike maximal p-dimensional submanifolds in the pseudo-hyperbolic space of signature (p,q). In the second part of the talk, I will discuss applications of this result in Teichmüller theory and for the study of Anosov representations. This is joint work with Graham Smith and Jérémy Toulisse.

In this talk, I will explain why minimal surfaces in spheres are very important for the theory of the Willmore functional. They serve as competitors, and those with low area are good candidates for minimzers of the Willmore functional among surfaces of a fixed genus. The picture in the higher codimension case is very incomplete. Lawson's bipolar surfaces in the 4- and 5-sphere are among the rare known examples of minimal surfaces that do not lie in a 3-sphere. I will explain the construction of the bipolar surface, I will emphasize a specific Klein bottle among that family and I will tell you about a recent result with Melanie Rothe in which we determine topological and embeddedness properties of the bipolar surfaces of Lawson's $\xi$- and $\eta$-family. This talk is based on work with Melanie Rothe, Patrick Breuning and Jonas Hirsch.

In this talk, no $\mathsf{G}_2$ background is assumed and all relevant $\mathsf{G}_2$ terminology will be defined. We discuss non-abelian Hodge theory on the punctured sphere for the split real Lie group $\mathsf{G}_2'$. We study almost-complex curves $\nu: \mathbb{C} \rightarrow \mathbb{S}^{2,4}$ in the pseudosphere $\mathbb{S}^{2,4}$ associated to polynomial sextic differential $q$ in the complex plane. Focusing on the asymptotic geometry, we detect stable regions and Stokes lines where the limits of $\nu$ change. Moreover, we find such polynomial almost-complex curves have polygonal boundaries in $\mathsf{Ein}^{2,3}$ satisfying a condition we call the annihilator property. Time permitting, we discuss a conjectural homeomorphism from a moduli space of polynomial sextic differentials to a moduli space of annihilator polygons in $\mathsf{Ein}^{2,3}$.

Hitchin’s theory of Higgs bundles associated holomorphic differentials on a Riemann surface to representations of the fundamental group of the surface into a Lie group. We study the geometry common to representations whose associated holomorphic differentials lie on a ray. In the setting of SL(3,R), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. This is joint work with John Loftin and Mike Wolf.