In this talk we present a result about a classification of the rotationally-symmetric solutions to an overdetermined problem in the 2-sphere. As a consequence, we give a rigidity result about a certain type of minimal surfaces and as a corollary of this result we provide a new characterization of the critical catenoid among the embedded free boundary minimal annulus in the unit ball. This is based in a join work with José M. Espinar.

In this talk we are going to discuss about new examples of translating solitons with entropy five and six.

The so-called Schiffer conjecture was stated by S.T. Yau in his famous list of open problems as follows: If a nonconstant Neumann eigenfunction $u$ of the Laplacian on a smooth bounded domain in $\mathbb{R}^2$ is constant on the boundary, then the domain is a disk. In this talk we will consider a version of such question for domains with disconnected boundary. Specifically, we consider Neumann eigenfunctions that are locally constant on the boundary, and we wonder if the domain has to be necessarily a disk or an annulus. We will show that the answer to the above question is negative. Indeed, there are nonradial Neumann eigenfunctions which are locally constant on the boundary of the domain. The proof uses a local bifurcation argument together with a reformulation of the problem by Fall, Minlend and Weth that avoids a problem of loss of derivatives. This is joint work with A. Enciso, A. J. Fernández and P. Sicbaldi.

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 this: 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 w/ Impera \& Rimoldi.) Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (joint w/ Gama \& Martín), using Morse theory and nodal set techniques, which extend Chini's classification.

In a recent work with A.Alarcón and I.Castro-Infantes we show that every open Riemann surface admits a complete conformal CMC-1 (constant mean curvature one) immersion in the three dimensional hyperbolic space. In this talk I aim to explain the main ideas in the proof of this result, which relies on the holomorphic representation of CMC-1 surfaces given by Robert Bryant in 1987, and applies modern complex analysis techniques.

In this talk, we investigate a pseudo-Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$. By using pseudo-anti commuting Ricci tensor, we give a complete classification of Hopf pseudo-Ricci-Bourguignon soliton real hypersurfaces in $G_2({\mathbb C}^{m+2})$ . Moreover, we have proved that there exists a non-trivial classification of gradient pseudo-Ricci-Bourguignon soliton $(M, {\xi}, {\eta}, {\Omega}, {\theta}, {\gamma}, g)$ on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$. In the class of contact hypersurface in $G_2({\mathbb C}^{m+2})$, we prove that there does not exist a gradient pseudo-Ricci-Bourguignon soliton in $G_2({\mathbb C}^{m+2})$

In this talk, we give a complete classification of Yamabe soliton, Ricci-Bourguignon soliton on real hypersurfaces in complex hyperbolic quadric ${Q^m}^*$ and complex hyperbolic space $CH^m$. Next as applications, we also give a complete classification of gradient Yamabe soliton and gradient Ricci-Bourguignon soliton on real hypersurfaces in complex hyperbolic quadric ${Q^m}^*$ and complex hyperbolic space $CH^m$. Related to this topics, finally we want to mention soliton problems and Fisher-Marsden conjecture in other Hermitian symmetric spaces like complex two-plane Grassmannians, complex hyperbolic two-plane Grassmannians, complex quadric or etc.

A smooth function $f$ on a Riemannian manifold $\widetilde{M}$ is isoparametric if $|\nabla f|$ and $\Delta f$ are constant on the level sets of $f$. A hypersurface $M \subset \widetilde{M}$ is an isoparametric hypersurface if it is a regular level set of an isoparametric function defined on $\widetilde{M}$. When the ambient space is a space form $\mathbb{Q}^{n}_c$, the definition of isoparametric hypersurface is equivalent to saying that the hypersurface has constant principal curvatures. However, in arbitrary ambient spaces of nonconstant sectional curvature, the equivalence between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures may no longer be true. In this talk, it will be presented characterization and classification results on isoparametric hypersurfaces with constant principal curvatures in the product spaces $ \mathbb{Q}_{c_{1}}^2 \times \mathbb{Q}_{c_{2}}^2$, for $c_{i} \in \{-1,0,1\}$ and $c_1 \neq c_2$. The talk is based on a joint work with Jo$\tilde{\rm a}$o Batista Marques dos Santos.

The moduli space of solutions to Nahm’s equations with values in the Lie algebra of a Lie group G (more generally, self-dual Yang-Mills equations) carries a complete hyperkähler structure, obtained via infinite-dimensional reduction by Hitchin (1987). Kronheimer (1989) proved that this is diffeomorphic to the total space of the cotangent bundle T*Gc of a complex Lie group. Both the hyperkähler structure on the moduli space and the diffeomorphism with T*Gc are proved to exist abstractly; hence, the resulting hyperkähler metric on T*Gc is challenging to describe explicitly even for basic Lie groups. We present joint work with Richard Melrose and Michael Singer obtaining the asymptotics of the metric on the moduli space and the resolution of the critical set of the Nahm vector. We also present an explicit description of the diffeomorphism and induced metric on T* in the case of G=SU(2).

One of the most extensively studied topics in Differential Geometry is minimal surfaces. This subject has connections with a wide range of mathematics areas, such as complex analysis, partial differential equations and topology. A classic problem in this context is the Plateau problem and recently much attention has been given to problems with the free boundary condition. In this talk, we will present some results about classification for constant mean curvature (CMC) surfaces with the free boundary condition in certain spaces. This is a joint work with Ezequiel Barbosa (UFMG) and Edno Pereira (UFSJ).

In 1841 Delaunay characterized surfaces of constant mean curvature $H=1$ in Euclidean 3-space invariant under rotation. The result was generalized by several authors to screw-motion invariant CMC surfaces in $\mathbb{E}(\kappa,\tau)$. In this more general setting CMC tubes can arise in addition to the Delaunay surfaces. In this talk I want to present existence conditions and talk about further properties of these tubes such as embeddedness and foliation.

In this talk, we will consider an arbitrary orientable Riemannian surface $M$ and an open relatively compact domain $\Omega\subset M$ with piecewise regular boundary. Given a Killing submersion $\pi:\mathbb{E}\to M$, we will discuss some properties of the divergence lines spanned by a sequence of minimal graphs over $\Omega$, as well as how they produce certain laminations in $\pi^{-1}(\Omega)$ whose leaves are vertical surfaces (after considering a subsequence). We will apply these results to give a general solution to the Jenkins-Serrin problem over $\Omega$ under natural necessary assumptions. This is a joint work with Andrea del Prete and Barbara Nelli.

Picone identity named after Mauro Picone (1885-1977) is classical in the theory of homogeneous linear second order differential equations yielding many results. In this talk, I will present a new generalised variable exponent Picone type identity for horizontal $p(x)$-Laplacian of general vector fields. The identity generalises several known results in literature. Then as an application, we will study the indefinite weighted Dirichlet eigenvalue problem for horizontal $p(x)$-sub-Laplacian on smooth manifolds and discuss some properties of the first eigenvalue and its corresponding eigenfunctions such as uniqueness, simplicity, monotonicity and isolation in the context of variable exponent Sobolev spaces. Further applications also yield Hardy type inequalities and Caccioppoli estimates with variable exponent.

A map is said to be homotopic equivalent to another map if there is a continuous path of maps connecting the two. Surprisingly a geometric flow such as the mean curvature flow provides natural paths to deform a map to a canonical representative in its homotopy class. I shall discuss this approach and its applications, and in particular some recent results regarding the homotopy class of maps between complex projective spaces of different dimensions. This is based on joint work with Chun-Jun Tsai and Mao-Pei Tsui.

We prove that given a finite set $E$ in a bordered Riemann surface $\mathcal{R}$, there is a continuous map $h\colon \overline{\mathcal{R}}\setminus E\to\mathbb{C}^n$ ($n\geq 2$) such that $h|_{\mathcal{R}\setminus E} \colon \mathcal{R}\setminus E\to\mathbb{C}^n$ is a complete holomorphic immersion (embedding if $n\geq 3$) which is meromorphic on $\mathcal{R}$ and has effective poles at all points in $E$, and $h|_{b\overline{\mathcal{R}}} \colon b\overline{\mathcal{R}}\to\mathbb{C}^n$ is a topological embedding. In particular, $h(b\overline{\mathcal{R}})$ consists of the union of finitely many pairwise disjoint Jordan curves which we ensure to be of Hausdorff dimension one. We establish a more general result including uniform approximation and interpolation.