Doctoral information ^*

Research topic // Tema de cercetare

Special type of submanifolds in (semi)-Riemannian manifolds

Short description:
We have in mind two distinct research directions:

I. Magnetic curves, as critical points of the Landau Hall functional, 
are intensively studied in the last years. Our target is to extend the 
notion of magnetic curve to maps between Riemannian manifolds, when
a magnetic field is defined on the target manifold. In such a way,
we define the notion of magnetic map and our objective is to study
geometric properties for these magnetic maps.

II. The main objects of this line of research are the CR-submanifolds 
(in the sense of Bejancu) satisfying certain natural geometric conditions.



Clase speciale de subvarietati in varietati (semi-) Riemanniene

Descriere:
Se vor avea in vedere doua directii de cercetare:

I. Curbele magnetice, ca puncte critice ale functionalei Landau Hall,
sunt intens studiate in ultima perioada. Se urmareste, de asemenea, extinderea
notiunii de curba magnetica la aplicatii intre varietati Riemanniene,
cand pe codomeniu este prezent un camp magnetic. Apare astfel
notiunea de aplicatie magnetica pentru care se vor studia diverse
proprietati geometrice remarcabile.

II. Cercetarea va avea ca tema principala studiul subvarietatilor CR
care satisfac anumite conditii geometrice naturale.

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Solitoni ai fluxului de curbură medie

Fluxul de curbură medie (MCF) a fost investigat pentru prima dată in 1978 de Brakke,
ulterior (in 1984) de Huisken. De atunci, subiectul a cunoscut o dezvoltare semnificativă atat
in ​​teoria masurii geometrice, cat ai an geometria Riemanniana, de exemplu, de Martin, dos Santos,
Savas-Halilaj, Smoczyk, Lopez, Rocha, Pipoli si multi altii. Ideea principala este de a considera
o familie neteda de imersii F_t : M --> N a unei varietati M netede, conexe (n-1)-dimensionale
intr-o varietate Riemanniani n-dimensionali N si un camp vectorial Killing pe N. F_t este o solutie
a fluxului de curbura medie pe (0, T ), T > 0, daca

d/dt F_t = − H \nu pe M x (0,T)
și
F_0 = f pe M,

unde f : M -> N este o hipersuprafata (initiala) data M_0. Aici, H reprezinta curbura medie a lui
F_t(M) in raport cu campul vectorial unitar normal \nu pe F_t(M).

O problema interesanta este cunoasterea evolutiei hipersuprafetei prin flux.
In functie de hipersuprafata initiala M_0, aceasta evolutie poate fi definita pentru
orice t > 0 sau se poate intampla ca fluxul să se "prabuseasca" la un moment finit.

Scopul este studiul translatorilor MCV in anumite spatii, adica hipersuprafete care evolueaza
prin translatii obtinute din grupul 1-parametric de izometrii generate de X si pe care le numim
solutii soliton ale MCV. Investigatia va fi dezvoltata si in codimensiune mai mare.

Poate ca exemplele de baza sunt date de curba grim reaper si de curba Yin-yang atunci cand
N este planul euclidian (2-dimensional) si evolutia este considerata de-a lungul unui vector
constant, respectiv in raport cu grupul de rotatie. Ne așteptam să obtinem mai multe exemple noi
in diverse spatii ambient.



Solitons of the mean curvature flow


The mean curvature flow (MCF) has first been investigated 1978 by Brakke, later (in 1984) by Huisken.
Since then, the topic had a great development both in geometric measure theory as well as in Riemannian
geometry, e.g. by Martın, dos Santos, Savas-Halilaj, Smoczyk, Lopez, Rocha, Pipoli and many others.
The main idea is to consider a smooth family of immersions F_t:M->N of a smooth, connected (n-1)-dimensional
manifold M into a n-dimensional Riemannian manifold N and a Killing vector field on N. F_t is a solution
of the mean curvature flow on (0, T ), T > 0, if

d/dt F_t = − H \nu on M x (0,T)
and
F_0 = f on M,

where f : M -> N is a given initial hypersurface M_0. Here, H denotes the mean curvature of
F_t(M) with respect to the normal unit vector field \nu on F_t(M).

An interesting question is to know the evolution of the hypersurface through the flow.
Depending on the initial hypersurface M_0, this evolution can be defined for all t > 0,
or it may occur that the flow collapses at some finite time.

The aim is the study of translators of MCV in some spaces, that is hypersurfaces that evolve
by translations obtained from the 1-parametric group of isometries generated by X and we also
call them soliton solution of the MCV. The investigation will be developed also in higher codimension.

Maybe the basic examples are given by the grim reaper and the Yin-yang curve when N is
the (2-dimensional) Euclidean plane and the evolution is considered along a constant vector
and with respect to the rotation group, respectively.
We expect to obtain several new examples in several ambient spaces.

(*) On May 20, 2015, I defended my habilitation thesis entitled:
Recent developments on magnetic curves in (pseudo)-Riemannian manifolds

Atestat de abillitare: OM 4718/11.08.2015
Afiliere la scoala doctorala de matematica: Hotarare Senat n.5 / 24.09.2015