Alcides de Carvalho Júnior
Index estimates of harmonic Gauss maps
Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within the Lie algebra of $\mathbb{G}$.
We prove that the energy index of the Gauss map of $\Sigma$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.

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Allan George de Carvalho Freitas
Rigidity for Serrin’s Problem in Riemannian Manifolds
In this lecture, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach Serrin’s problem in bounded domains of manifolds endowed with a conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. This lecture joint works with A. Roncoroni (Politecnico di Milano, Italy), M. Santos (UFPB, Brazil), M. Andrade (UFS, Brazil) and Diego Marín (Universidad de Granada, Spain).

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Almir Rogério Silva Santos
Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures
Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $\lambda$ and $\mu$, there exist at least two distinct conformal metrics $g_{\lambda,\mu}=e^{2u_{\mu,\lambda}}g$ and $g^{\lambda,\mu}=e^{2u^{\mu,\lambda}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + \mu$ and $h + \lambda,$ respectively. Additionally, we employ the method used by Borer et al. (2015) to study the blowing up behavior of the large solution $u^{\mu,\lambda}$ when $\mu\downarrow 0$ and $\lambda\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles. This is joint work with R. Caju (University of Chile) and T. Cruz (UFAL).

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Ari João Aiolfi
The Dirichlet problem for the minimal surface equation on G-invariant domains
In our talk we will present some recent results relative to the Dirichlet problem for the minimal surface equation on G-invariant domains of a complete Riemannian manifold M, for G-invariant boundary data, where G is a Lie subgroup of the group of isometry of M that acts freely and properly on M. The results contemplate unbounded domains which are not exterior domains.

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Cicero Tiarlos Nogueira Cruz
On static manifolds satisfying an overdetermined Robin type condition on the boundary
In this talk, we consider static manifolds with nonempty boundary. In this case, we suppose that the potential function also satisfies an overdetermined Robin type condition on the boundary. We prove a rigidity theorem for the Euclidean closed unit ball. More precisely, we give a sharp upper bound for the area of the zero set of the potential V, when it is connected and intersects the boundary. We also consider the case where the boundary does not intersect the boundary. This is a joint work with Ivaldo Nunes.

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Eraldo Almeida Lima Júnior
Characterization of noncompact Trapped, r-Trapped and Totally Trapped Submanifolds in Spacetimes
In his seminal paper Penrose related the trapped surfaces and the so called Black
Holes which are regions in space where the even the light cannot scape. Recently many characterizations were obtained for compact trapped surfaces and
submanifolds in order to understand Black Holes in spacetimes of usual or higher
dimensions. Based on that we studied noncompact trapped, r-trapped and the
totally trapped submanifolds in Generalized Robertson Walker, Standard Static
Spacetimes, Schwarzschild type among others spacetimes. In this talk we will
present characterizations of parabolic trapped and r-trapped submanifolds and
complete totally( super-symmetry) trapped submanifolds. Moreover we will
present examples justifying our hypothesis and its realations to Physic

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Fábio Reis dos Santos, Maximal spacelike submanifolds in the indefinite product space, In this talk, we deal with maximal spacelike submanifolds into the indefinite product space $M^{n}{q}(c)\times\mathbb{R}{\delta}$, where $M^{n}{q}(c)$ represent an indefinite manifold with constant sectional curvature $c\in{-1,1}$ with index $q$, and $\mathbb{R}{\delta}$ denotes the real line endowed with the Euclidean metric if $\delta=1$ or the Lorentzian metric, if $\delta=-1$. In this setting, we establish a Simons-Calabi formula for maximal submanifolds immersed into the indefinite product space. And, in the end, we presente some applications.

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Flávio França Cruz, An extension of Liebmann’s Theorem, The classical Liebmann’s Theorem asserts that a compact connected convex surface in R3 with constant mean curvature (CMC) is a totally umbilical sphere. In this presentation, we introduce an extension to Liebmann’s Theo- rem, focusing on surfaces with boundaries. Specifically, we demonstrate that a locally convex, embedded compact connected CMC surface, bounded by a convex curve, lives in a half space of R3. In particular, we conclude that sphe- rical caps are the only locally convex, embedded compact connected nonzero CMC surface bounded by a circle.

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Flávio Almeida Lemos, Second eigenvalue of the CR Yamabe operator, Abstract
Let M be a compact, connected, strictly pseudoconvex CR manifold. We give some properties of the CR Yamabe Operator . We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

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Gisele Teixeira Paula, The stretch of a semi-arithmetic Fuchsian group., Semi-arithmetic Fuchsian groups is a wide class of discrete groups of isometries of the hyperbolic plane which includes arithmetic Fuchsian groups, hyperbolic triangle groups, groups admitting a modular embedding, and others. Most of my talk will foccus on the definition of the stretch of a semi-arithmetic group, which is a new geometric invariant of the commensurability class of those groups. We introduce this invariant in order to generalize the notion of groups admitting modular embeddings, which is an important class of semi-arithmetic groups. The definition of stretch uses Lipschitz geometry and the proof of its invariance with respect to commensurability is based on the notion of the Riemannian center of mass developed by Karcher and collaborators. This is a joint work with M. Belolipetsky, G. Cosac and C. Dória.

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Nikolai Goussevskii, Constracting complex hyperbolic structures on disc bundles over non-orientable surfaces, n this work, we construct first examples of complex hyperbolic structures on disc bundles over closed non-orientable surfaces. To do this, we constract discrete faithful representations of the fundamental groups of closed non-orientable surfaces in the isometry group of complex hyperbolic space of dimension 2.

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Henrique Amador Puel Martins, Transverse geometry of Lorentzian foliations, We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking–Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds.

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HIURI FELLIPE SANTOS DOS REIS, SOLITON SOLUTIONS TO THE CURVE SHORTENING FLOW ON THE SPHERE, It is shown that a curve on the unit sphere is a soliton solution to
the curve shortening flow if and only if its geodesic curvature is proportional to
the inner product between its tangent vector and a fixed vector of R3. Using
this characterization, we describe the geometry of such a curve on the sphere,
we study its qualitative behavior, and we prove the convergence of the curve
to the equator orthogonal to the fixed vector

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Iury Domingos, Immersed Ricci surfaces, Please find the abstract attached.

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Leandro de Freitas Pessoa, L^p-parabolicity on Riemannian manifolds, We introduce the concept of L^p-parabolicity for Riemannian manifolds. This concept, such as the classical parabolicity (case p=1), is defined in terms of a nonlinear capacity and is characterized via L^q-Liouville property for positive superharmonic functions, where p and q are Hölder conjugate exponents. We also discuss on sufficient sharp volume growth conditions for the validity of the L^p-parabolicity of a complete Riemannian manifold. This is a joint work with Alexander Grigor’yan and Alberto G. Setti.

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Luciana Aparecida Alves, Decompositions on Flag Bundles and Uniform Bounds of Cocycle

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Luiz Hartmann, Curvature estimates for graphs in warped product spaces , We prove local and global upper estimates for the infimum of the mean curvature,
the scalar curvature and the norm of the shape operator of graphs in a warped
product space. Using these estimates, we obtain some results on pseudo-hyperbolic
spaces and space forms. This was a joint work with A. Barreto (UFSCar) and F. Coswosck (UFES).

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Matheus Hudson Gama dos Santos, Mean curvature flow in an extended Ricci flow background, In this talk, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying mean curvature flow in a Ricci flow background. Mainly, the functional we focus on the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties, from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman’s modified extended Ricci flow. In this time-derivative formula an extension of Hamilton’s differential Harnack expression on the boundary integrand appears. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish Huisken’s monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Finally, we present examples of mean curvature solitons in an extended Ricci flow background.

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Maria Rosilene Barroso dos Santos, A Simons type formula for spacelike submanifolds in semi-Riemannian warped product and its applications, In this talk, we present a formula for the Laplacian of squared norm of the second form fundamental for a spacelike submanifold in a Semi-Riemannian Warped product and then we give some applications. In the special case of ambient space to be the basic cosmological models: Lorentz-Minkowski, de Sitter, anti-de Sitter, and Einstein-de Sitter spacetimes, we characterize precisely a class of submanifolds with zero mean curvature vector field.

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Neiton Pereira da Silva, Geometry of invariant almost Hermitian submanifolds of flag manifolds, Resumo em anexo.

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Newton Mayer Solórzano Chávez, Minimal surfaces in non-Minkowskian Randers spaces, In this work we investigate minimal hypersurfaces in R^n with respect to Busemann-Hausdorff measure in a class of Finsler n-spaces (R^n,F_b=\alpha+\beta), called Randers spaces, where \alpha is the Euclidean metric and \beta is a one form. We particularly examine graphs defined on the xy-plane that are invariant under one-dimensional isometry groups of (R^3,F_b). By reducing the minimal graph equation to an ordinary differential equation (ODE), we obtain a new class of explicit examples of minimal surfaces in Finsler Geometry.

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Nicolau Aiex, Quantitative Estimates on the Singular Set of Minimal Hypersurfaces, The work I want to present is about the structure of the singular set of minimal hypersurfaces in higher dimensions. We prove an estimate on the Minkowski measure of the singular set to prove a bound on the Minowski dimension of the singular set, which is a strictly stronger result than the classical Hausdorff estimates. Please find a more precise abstract in the file attached.

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Rayssa Caju, Moduli Space theory for complete, constant Q-curvature metrics, The Q- curvature equation, a fourth-order elliptic PDE with a critical exponent, is a conformal equation that has drawn considerable attention because of its relation with a natural concept of curvature. In this talk, our main goal is to study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real algebraic variety with formal dimension equal to the number of punctures. 

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Rondinelle Marcolino Batista, Rigidity of free boundary minimal disks in mean convex three-manifolds, In this lecture, we presente a local rigidity result for free boundary minimal two-disk \Sigma that locally maximize the modified Hawking mass on a Riemannian three manifold M with positive lower bound on its scalar curvature and mean convex boundary. Assuming strict stability of \Sigma, we prove that a neighborhood of it in M^3 is isometric to one of the half de Sitter Schwarzschild metric. This is a joint work with Barnabé Lima (UFPI) and João Silva (UFPI).

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Ronaldo Freire de Lima, SOLITONS TO MEAN CURVATURE FLOW IN THE HYPERBOLIC 3-SPACE, We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic 3-space H3, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in H3, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in H3 and apply it to prove that properly immersed translators to MCF in H3 are
not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the x3-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in H3 which have constant mean curvature.

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Roney Santos, On the stability of free boundary minimal submanifolds in conformal balls, A classical and interesting research subject in differential geometry are submanifolds of a given Riemannian manifold that are critical points of a variational problem for some functional. In order to decide if this critical point is a local maximum or minimum, it is natural to look to the second derivative of the considered functional.Our goal in this talk is to present some spaces that does not admit free boundary minimal submanifolds, which are critical points of the volume functional. In particular, we want to talk about a recent work joint with Alcides de Carvalho when we show that a conformally flat manifold that is compact, simply-connected, $\delta$-pinched and has convex boundary does not admit free boundary minimal submanifold provided that $2 \leq k \leq n-\delta^{-1}$ and $k \leq n-2.$

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Rodrigo Francisco dos Santos , Gravity Equivalent to Teleparalelism: Some Basic Geometrical Aspects, We review the book of Ruben Aldrovandi and Jose Geraldo Pereira about Teleparallel Gravity. Teleparallel Gravity is an alternative the General Relativity for description the gravitational interaction. The diference between General Relativity and Teleparallel Gravity be with fact the General Relativity associated the curvature to the gravitational interaction, in the Teleparallel Gravity the mathemathical object associated the gravitational field is the torsion, the field is called Weintzembock Conection. We discuss briefly the advantages of Teleparallel Gravity front General Relativity. And Weintzenbock Conection proprieties .

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Stefano Nardulli, AN ALLARD-TYPE BOUNDARY REGULARITY THEOREM FOR 2d MINIMIZING CURRENTS IN ARBITRARY CODIMENSION AT SMOOTH CURVES WITH ARBITRARY MULTIPLICITY, This presentation delves into geometric measure theory, in particular into the boundary regularity theory realm of integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q\llbracket\Gamma\rrbracket$, where $Q\in \mathbb N \setminus {0}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$ is a point where the density of $T$ is strictly below $\frac{Q+1}{2}$, then the current is regular at $q$. The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$. As a corollary, if $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q\llbracket\Gamma\rrbracket$ is regular in a neighborhood of $\Gamma$. This is a joint work with Camillo De Lellis and Simone Steinbr\”uchel published in \cite{AA}.

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Sylvia Ferreira da Silva, The total mean curvature functional for H-surfaces in product spaces, In this work we present some results concerning of total mean curvature functional $\mathcal{H}$, for submanifolds into the Riemannian product space $\mathbb{S}^{n}\times\mathbb{R}$. We present its first variational formula and define two second order differential operators followed by a nice integral inequality relating both of them. Our main result establishes an integral inequality for closed stationary $\mathcal{H}$-surfaces in $\mathbb{S}^{n}\times\mathbb{R}$, characterizing the cases where the equality is attained.

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Ruy Tojeiro de Figueiredo Junior, Submanifolds with constant Moebius curvature and flat normal bundle, We classify isometric immersions $f:M^{n}\to \mathbb{R}^{n+p}$, $n \geq 5$ and $2p \le n$, with constant Moebius curvature and flat normal bundle.

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Dragomir Tsonev, THE ATIYAH-SINGER INDEX THEOREM – A GLIMPSE INTO THE DEEPNESS, The celebrated Atiyah-Singer Index theorem, without a shadow of a doubt, stands as one of the deepest results of mathematics ever discovered. Its beauty and grace naturally permeate some of the fundamental areas of mathematics. It generalises a variety of deep theorems and has far reaching applications. And who knows whether or not it will not manifest itself again in some newer context. Both its statement and its different proofs necessitate an interdisciplinary mathematics, ergo, much time and effort are needed to penetrate into the the essence and the details. Notwithstanding the latter difficulties, the goal of this short talk will be a humble attempt to bring forward, as much as possible, the geometry which lays behind this remarkable theorem. Hopefully this effort might instigate some interesting discussions among the participants of the XXI School Of Differential Geometry.

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Valter Borges
Warped product Ricci solitons and warping function estimates

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Willian Isao Tokura
Gradient Einstein-type warped metrics
We provide the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics. In this setting, we prove gradient estimates for warping functions by extending the approach in Li-Yau and Souplet-Zhang’s work. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics.