Banca de DEFESA: João Batista Marques dos Santos

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : João Batista Marques dos Santos
DATE: 28/04/2023
TIME: 10:00
LOCAL: PPGMAT
TITLE:

On isoparametric hypersurfaces in 4-dimensional product spaces

KEY WORDS:

isoparametric hypersurfaces, product spaces, parallel hypersurfaces, constant
principal curvatures, mean curvature flow.

PAGES: 76
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

In this work, we study isoparametric hypersurfaces in product manifolds of
dimension 4. First of all, we characterize and classify the isoparametric hypersurfaces with
constant principal curvatures in the product spaces Q c1 x Q c2 , where Q ci is a space form with
constant sectional curvature ci, for ci {-1,0,1} e c1 c2. We show that such hypersurfaces are
given as open subsets of either a product hypersurface, where one factor is a curve of
constant curvature, or a diagonal structure in H 2 x R 2 , constructed from horocycles in H 2 and
straight lines in R 2 .
Next, we classify the hypersurfaces in Q 3 x R with the three distinct constant principal
curvatures, where in this case . We show that such hypersurfaces are cylinders over

isoparametric surfaces of Q 3 with two non-null distinct principal curvatures. We also prove
that the hypersurfaces with constant principal curvatures in Q 3 x R are isoparametric.
Furthermore, we provide a necessary and sufficient condition for an isoparametric
hypersurface on Q 3 x R to have constant principal curvatures.
Finally, we describe the evolution by the mean curvature flow of isoparametric hypersurfaces
in product manifolds of dimension 4. We show that the evolution of isoparametric
hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a
reparametrization of the flow by parallel hypersurfaces in a short time, as long as the
uniqueness of the mean curvature flow holds for the initial data and the corresponding
ambient space. Through this result, we describe the evolution of the hypersurfaces classified
in the first and second parts of the work. We also describe the evolutions of isoparametric
hypersurfaces in S 2 x S 2 and H 2 x H 2 , classified by Urbano (2019) and Dong Gao, Hui Ma and
Zeke Yao (2022), respectively, and of isoparametric hypersurfaces in Q 3 x R with g distinct
constant principal curvatures, g {1,2}, classified by Chaves and Santos (2019).

BANKING MEMBERS:
Externo à Instituição - BENEDITO LEANDRO NETO - UFG
Externo à Instituição - MIGUEL DOMINGUEZ VAZQUEZ - USC
Presidente - 1984498 - JOAO PAULO DOS SANTOS
Interno - 3193644 - PEDRO ROITMAN
Interno - 3101411 - TARCISIO CASTRO SILVA