## Geometry

## Members

Prof. Abdênago Alves de Barros

Prof. Antonio Caminha Muniz Neto

Prof. Antonio Gervásio Colares

Prof. Ernani de Sousa Ribeiro Júnior

Prof. Florentiu Daniel Cibotaru

Prof. Frederico Vale Girão

Prof. Gregório Pacelli Feitosa Bessa

Prof. Jonatan Floriano da Silva

Prof. Jorge Herbert Soares de Lira

Prof. Levi Lopes de Lima

Prof. Marco Magliaro

Prof. Rafael Montezuma pinheiro Cabral

## Description

The Differential Geometry Research Group at UFC, one of the strongest in Brazil, congregates eleven professors of the Mathematics Department of UFC, eight of which hold CNPq’s research grants.

The group’s international standard of excellence shows itself when one glances at the wide range of subareas of Differential Geometry encompassed by the scientific interests of its members. Among these, we highlight Geometric Analysis, Geometric PDEs, the interplay between Geometry and Physics, Minimal Surface Theory, general Submanifold Geometry, Global Geometry and Index Theory. Such a standard is also shown by the long term collaboration of the group with researchers of several other renowed centers, here and abroad, as well as by the several dozens of our former M.Sc. and PhD students which enter the Mathematical Departments of various brazilian universities.

Below, you find the list of the members of the group, together with brief résumés of their scientific interests and research.

**Abdênago Barros**

Abstract:

Selected publications:

**Antonio** **Caminha**

Abstract: my primordial interests in Differential Geometry concentrate on the interplay of Geometric Analysis and the geometry of submanifolds of naturally reductive semi-Riemannian symmetric spaces. More specifically, my research activities have been centered on problems related to the uniqueness of submanifolds of such spaces, subjected to extrinsic curvature constraints, mostly when such problems are amenable to the methods of Geometric Analysis. To this end, the generic strategy is to apply and/or develop new versions of maximum principles for first or second order linear partial differential operators naturally attached to the submanifolds.

Selected publications:

[1] L. J. Alías, A. Caminha and F. Y. do Nascimento. *A maximum principle related to volume growth and applications. *Ann. Mat. P. Appl. 200 (2021), 1637-1650.

[2] L. J. Alías, A. Caminha and F. Y. do Nascimento.* A maximum principle at infinity with applications.* J. Math. Anal. Appl. 474 (2019), 242-247.

[3] F. Camargo, A. Caminha and H. F. de Lima. *Bernstein-type theorems in semi-Riemannian warped products*. Proc of the Amer. Math. Soc. 139 (2011), 1841-1850.

[4] A. Caminha. *The geometry of closed conformal vector fields on Riemannian spaces*. Bull. Braz. Math. Soc. New Series, 42 (2011), 277-300.

[5] A. Caminha. *A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds*. Diff. Geom. Appl. 24 (2006), 652-659.

**Ernani Ribeiro Jr.**

Abstract: my research activities circle around the investigation of the geometry of canonical metrics in smooth manifolds, i.e., Einstein metrics, quasi-Einstein metrics, critical metrics of Riemannian functionals and Ricci solitons. Such metrics naturally arise in the study of geometric flows, variational problems, differential equations and general relativity. In recent years, they have caught the attention of researchers in Geometric Analysis, mainly due to their several applications to Geometry and Physics. In this context, we aim at characterizing such metrics and constructing explicit examples which could allow us to understand their properties, as well as to obtain new applications. We address these general problems through a comprehensive approach combining tools from PDE and Differential Geometry.

Selected publications:

[1] H.-D. Cao, E. Ribeiro Jr. and D. Zhou. *Four-dimensional complete gradient shrinking Ricci solitons*. J. Reine Angew. Math. (Crelle’s Journal) (2021).

[2] X. Cheng, E. Ribeiro Jr. and D. Zhou. *Volume growth estimates for Ricci solitons and quasi-Einstein manifolds*. J. Geom. Analysis (2021).

[3] R. Batista, E. Ribeiro Jr. and M. Ranieri. *Remarks on complete noncompact Einstein warped products*. Comm. Anal. Geom. 28 (2020), 549-565.

[4] R. Batista, R. Diógenes, M. Ranieri and E. Ribeiro Jr. *Critical metrics of the volume functional on compact three-manifolds with smooth boundary.* J. Geom. Anal. 27 (2017), 1530-1547.

[5] A. Barros, R. Diógenes and Ribeiro Jr.* Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary.* J. Geom. Anal. 25 (2015), 2698-2715.

**Florentiu Daniel Cibotaru**

Abstract: Poincaré’s duality, in its various incarnations, is one of the pillars of classical Algebraic and Differential Topology. Starting with the seminal work of R. Harvey and B. Lawson in the nineties, we have been seeing a renewed interest in the search for manifestations and refinements of such duality, which involves the usage of tools as currents, geometric measure theory, Morse theory and complex or real analytic geometric. The main purpose is to use these investigations to either refine or generalize, in a “singular” context, classical results from Differential Geometry or Index Theory.

Selected publications:

[1] D. Cibotaru and W. Pereira. *Non-tame Morse-Smale flows and odd Chern-Weil theory. *Can. J. Math. (2021), 1-38, DOI 10.4153/S0008414X21000353.

[2] D. Cibotaru and S. Moroianu. *Odd Pfaffian forms*. Bull. Braz. Math. Soc. (2021), DOI 10.1007/s00574-020-00239-0.

[3] D. Cibotaru. *Bioriented flags and resolutions of Schubert varieties*. Math. Nach. 293 (2020), 449-474.

[4] D. Cibotaru. *Chern–Gauss–Bonnet and Lefschetz duality from a currential point of view. *Adv. Math. 317 (2017), 718–757.

[5] D. Cibotaru. *Vertical flows and a general currential homotopy formula*. Indiana Univ. Math. J. 65 (2016), 93-169.

**Frederico Girão**

Abstract: several are the geometric quantities attached to a hypersurface *M* of a Riemannian manifold *N*, say, the area of *M*, the volume that *M* encloses, the total mean curvature of *M* and the mass of *M* (when *M* is asymptotically Euclidean or hyperbolic). My research focuses on inequalities involving such quantities, the so-called geometric inequalities, when the ambient Riemannian manifold is static. These inequalities include Alexandrov-Fenchel-type inequalities, as well as positive mass and Penrose-type inequalities (when *M* is asymptotically Euclidean or hyperbolic).

Selected publications:

[1] F. Girão, D. Pinheiro, N. M. Pinheiro and D. Rodrigues. *Weighted Alexandrov-Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang and Wu*. Amer. Math. Soc. Proc. 149 (2021), 369-382.

[2] F. Girão and D. Rodrigues. *Weighted geometric inequalities for hypersurfaces in sub-static manifolds*. Bull. London Math. Soc. 52 (2020), 121-136.

[3] A. de Sousa and F. Girão. *The Gauss-Bonnet-Chern mass of higher-codimension graphs*. Pacific J. Math. 298 (2019), 201-216.

[4] L. L. de Lima and F. Girão. *An Alexandrov-Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality*. Ann. H. Poincaré 17 (2016), 979-1002.

[5] L. L. de Lima and F. Girão. *The ADM mass of asymptotically flat hypersurfaces*. Trans. Amer. Math. Soc. (Online) 367 (2015), 6247-6266.

**Gregório Pacelli Bessa**

Abstract:

Selected publications:

**Jonatan Silva**

Abstract: my research concentrates on the study of the properties of the mean curvatures of immersed hypersurfaces of Riemannian manifolds, as well as of spacelike hypersurfaces of Lorentz manifolds, focusing on classification problems. More recently, I have been working on the problem of stability of hypersurfaces with anisotropic structure (i.e, with anisotropic Weingarten operator) of constant curvature ambient spaces, as well as in the problem of classification of complete maximal spacelike hypersurfaces of the anti-de Sitter space having either constant scalar or Gauss-Kronecker curvature.

Selected publications:

[1] M. A. L. Velásquez, A. F. A. Ramalho, J. F. da Silva and J. Q. Oliveira. *Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products*. Nonlinear Anal. – Theory, Methods and Appl. 197 (2020), 111865.

[2] M. A. L. Velásquez, H. F. de Lima, J. F. da Silva and A. M. Oliveira. *Stable compact spacelike hypersurfaces in the de Sitter space as maxima of a linear combination of area and volume*. Manuscripta Math. 159 (2019), 229-245.

[3] J. F. Silva, H. F. de Lima and M. A. L. Velásquez. *Rigidity of complete hypersurfaces in the Euclidean space via anisotropic mean curvatures*. Bull. Braz. Math. Soc. (Online), 47 (2016), 971-987.

[4] A. G. Colares and J. F. Silva. *Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume*. Math. Z. 275 (2013), 595-623.

**Jorge Herbert Lira**

Abstract: lying in the interface of Geometric Analysis and Geometric PDEs, my research makes use of the apparatus of these subareas of Differential Geometry to model and solve geometric and physical problems in curved mutidimensional spaces. More precisely, recurrent themes in my research are the study of elliptic and parabolic PDEs whose solutions locally describe surfaces with prescribed extrinsic curvature. Some examples of these geometric objects I have been investigating are surfaces of (linear and nonlinear) constant mean curvatures, capillary surfaces, solitons of the mean curvature flow and, more recently, critical surfaces for fourth order energies. Another research direction I have been following, also related to Geometric Analysis and Geometric PDEs, is the study of constraint conditions for the field equations of General Relativity. In this effort, I have been continuously supervising PhD candidates, Post-Doc fellows and collaborating with other researchers from Brazil, Spain, France, Italy and the USA.

Selected publications:

[1] J. Lira, L. Alías, M. Rigoli. *Geometric elliptic functionals and mean curvature*. Ann. Sc. Norm. Sup. Pisa – Classe di scienze 15 (2016), 609-655.

[2] J. L. Barbosa, J. Lira, V.Oliker. *Closed weingarten hypersurfaces in warped product manifolds*. Indiana Univ. Math. J. 58 (2009), 1691-1718.

[3] M. Dajczer, J. Lira. *Killing graphs with prescribed mean curvature*. Calc. Var. PDE 33 (2008), 231-248.

[4] L. Alías, J. Lira, M. Malacarne. *Constant higher-order mean curvature hypersurfaces in riemannian spaces*. J. Inst. Math. Jussieu 5 (2006), 527-562.

[5] D. Hoffman, J. Lira, H. Rosenberg. *Constant mean curvature surfaces in M × R*. Trans. Amer. Math. Soc. 358 (2006), 491-507.

**Levi Lima**

Abstract: I have been working on some themes centered on global questions of Geometric Analysis having a strong Mathematical Physics motivation. More precisely, in the last decade my research interests have been focused in the mathematical aspects of problems of General Relativity (Penrose and mass inequalities for a large class of initial data sets), Stochastic Analysis (Feynman-Kac formulas for the heat semigroups of generalized Dirac Laplacians, under mixed boundary conditions) and Index Theory (conformal, applied to detect obstructions to the existence of positive scalar curvature metrics in certain noncompact manifolds).

Selected publications:

[1] L. L. de Lima e F. Girão. *An Alexandrov-Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality*. Ann. H. Poincaré 17 (2016), 979-1002.

[2] S. Almaraz, E. Barbosa, L. L. de Lima. *A positive mass theorem for asymptotically flat manifolds with a non-compact boundary*. Comm. Anal. Geom. 24 (2016), 673-715.

[3] L. L. de Lima, P. Piccione, M. Zedda. *On bifurcation of solutions of the Yamabe problem in product manifolds*. Ann. Inst. H. Poincaré C, Analyse non linéaire 29 (2012), 261-277.

[4] J. X. da Cruz Neto, L. L. De Lima, P. R. Oliveira. *Geodesic algorithms in riemannian geometry*. Balkan J. Geom. Appl. 3 (1998), 89-100.

[5] L. L. de Lima, W. Rossman. *On the index of constant mean curvature 1 surfaces in hyperbolic space. *Indiana Univ. Math. J. 29 (1998), 685-723 29.

**Marco Magliaro**

Abstract:

Selected publications:

**Rafael Montezuma**

Abstract: my primary research interests are in Geometric Analysis, with a special emphasis on variational methods and minimal surfaces. I also have interests in other subareas of Differential Geometry, such as geometric PDEs (mainly those involving variational techniques, such as eigenvalue problems in manifolds) and Morse theory. More precisely, my research has been oriented toward problems related to the min-max construction of minimal surfaces and the study of the objects thus produced. These methods can be briefly described as Morse theories for the functional area of surfaces of a tridimensional ambient manifold, or, more generally, for the volume of hypersurfaces of higher dimensional spaces. The involved objects and techniques are directly related and/or are motivated by developments in other subareas of Geometry, like Harmonic Mapping Theory, Willmore and CMC Surfaces, extremal eigenvalue problems, systolic inequalities, as well as connections with phase transition problems in PDEs.

Selected publications:

[1] R. Montezuma, F. C. Marques, A. Neves. *Morse inequalities for the area functional*. J. Diff. Geom. (2021). Accepted for publication.

[2] R. Montezuma. *On Free-Boundary Minimal Surfaces in the Riemannian Schwarzschild Manifold*. Bull. Braz. Math. Soc., New Series (2021). https://doi.org/10.1007/s00574-021-00245-w

[3] L. Ambrozio, R. Montezuma. *On the two-systole of real projective spaces*. Rev. Mat. Iberoam. 36 (2020), 1979-1988.

[4] R. Montezuma. *A mountain pass theorem for minimal hypersurfaces with fixed boundary*. Calc. Var. PDE 59 (2020), paper No. 188, 30 pp.

[5] R. Montezuma. *Min-max minimal hypersurfaces in non-compact manifolds*. J. Diff. Geom. 103 (2016), 475-519.