Banca de DEFESA: MATHEUS BARBOSA MARTINS

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : MATHEUS BARBOSA MARTINS
DATE: 25/11/2022
TIME: 14:00
LOCAL: Sala da Pós-Graduação, Bloco 12
TITLE:

Morse index estimates: results of lower bounds, classification, gaps, and rigidity.

KEY WORDS:

Minimal Hypersurface; Morse Index; Gap In The Index; Lower Bound; Weighted Spaces; Free Boundary.

PAGES: 1
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUBÁREA: Geometria e Topologia
SPECIALTY: Geometria Diferencial
SUMMARY:

In this Doctoral Thesis, we obtain several estimates for the Morse index of minimal hypersurfaces in five different settings. More precisely,

(1): We prove the nonexistence of a closed two-sided minimal hypersurface immersed in the real projective space RPn+1 with index two.

(2): We prove a gap of the Morse index of a orientable, closed minimal hypersurface immersed in a finite product of spheres. Such estimates are given as a function of the radii and dimensions of the spheres.

(3): We study orientable complete f-minimal free boundary hypersurfaces in a domain Ω of the weighted Euclidean space (R^{n+1}, g_{can}, e^{−f}dµ). In this case, we get lower bounds for the index by an affine function involving a topological quantity. If the hypersurface is compact, this quantity is its first Betti number.

(4): We consider operators of the type ∆_f + W − aK on surfaces in weighted Riemannian manifolds (M^3 , g, e^{−f}dµ), where W is a locally integrable function, K is the Gaussian curvature of the surface, and a is a positive integer. We obtain some results about the topology and volume growth of geodesic ball on f-stable constant weighted mean curvature surfaces. In addition, we also get a lower bound for the first eigenvalue of the stability operator.

(5): We obtain monotonicity and density formulas for hypersurfaces with a non-empty boundary, properly embedded in a warped product of type I ×_h S^2 . Finally, we present a method to calculate a region of stability for totally geodesic cones in a space conformal to warped products of the form I ×_h S^2 or I ×_h R^2 with curvature constant Ricci

BANKING MEMBERS:
Presidente - 2474631 - MARCIO HENRIQUE BATISTA DA SILVA
Interno(a) - 2346806 - CICERO TIARLOS NOGUEIRA CRUZ
Interno(a) - 1165266 - FELICIANO MARCILIO AGUIAR VITORIO
Externo(a) à Instituição - ALLAN GEORGE DE CARVALHO FREITAS - UFPB
Externo(a) à Instituição - IVALDO PAZ NUNES - UFMA