I made my Ph.D.  with Professor James Eells in 1987 at the University of Warwick, UK, supported by a fellowship from Calouste Gulbenkian Foundation, Lisbon, and made a post-doc at Maryland University, USA, with support of a JNICT fellowship. During my PhD I spent one  year (1986) at ICTP-International Centre for Theoretical Physics, Triste, Italy.

I work within Riemannian and pseudo-Riemannian Geometry and Geometric Analisys, namely submanifolds with constant mean curvature, isoperimetric inequalities and rigidity results, using maximum principles of PDE or evolution equations (mean curvature flow), and more recently I've  been working mainly in Spectral Geometry.

I've been reviewer for the Mathematical Reviews of the American Mathematical Society, and the Zentralblatt Math, and a referee for 15 international mathematical journals.

Most relevant works

Graphs with parallel mean curvature. Proceedings of the American Mathematical Society 107 (2) (1989), 449-449. 10.2307/2047835. (59 citations (Google source))

Graphic Bernstein results in curved pseudo-Riemannian manifolds, (with G. Li). Journal of Geometry and Physics 59 (9) (2009), 1306-1313.  10.1016/j.geomphys.2009.06.011 (22 citations)

Mean curvature flow of spacelike graphs, (with G. Li). Mathematische Zeitschrift 269, 3-4 (2011), 697-719. 10.1007/s00209-010-0768-4. (21 citations)

The isoperimetric problem in higher codimension, (with F. Morgan). Manuscripta Mathematica 142 (3-4) (2013): 369-382. 10.1007/s00229-012-0604-8.

Spherical symerization and the first eigenvalue of geodesic disks on manifolds, (with P. Freitas and J. Mao).  Calculus of Variations and Partial Differential Equations, 51 (2014), 701-724. 10.1007/s00526-013-0692-7. (25 citations)

Families of non-tiling domains satisfying Pólya's conjecture (with P. Freitas). Preprint (2022), submited. 

I was awarded with the Silver Medal of L'Association des Amis de Robert Schuman (1988).

homepage of Grupo de Física Matemática UL    
previous homepage http://cfif.ist.utl.pt/~isabel/

Participation in Research Projects

Member of the European Project Gadget (Global Analysis and Differential Geometry, European Team), in the framework of the UE programme Human Capital and Mobility, with the reference ERBCHRXCP920050, (1992-2002).

PI in the FCT project POCIPPCDT/MAT/60671/2004, Special Holonomy Geometry, (1/5/2005-3/4/2008).

Research member of the original team in the FCT project PTDC/MAT/101007/2008. Eigenvalues of the Laplacian: analytic, geometric and computational aspects, (18/01/2010-17/05/2013), with Freitas as the PI.

Research member in the FCT project PTDC/MAT/118682/2010, Geometrias Riemannianas Especiais, (01/01/2012-30/12/2014), with Rui Albuquerque as PI.

Collaborators

I' have colaborated with the following Mathematicians:

Pedro Freitas (U.Lisboa)
Frank Morgan (Williams College)
Jing Mao (Hubei U.)
Ana Cristina Ferreira (U.Minho)
Juan J. Salamanca (U.Córdoba)
Rui Albuquerque (U.Évora)
Ana Pereira do Vale (U.Minho)
Giorgio Valli (U.Pavia)
Guanghan Li (Hubei U.)
Duong Minh Duc (Ho-Chi-Minh City U.)
Marco Rigoli (U.Milano)
Jean-Paul Ezin (IMSP Porto-Novo, Benin)
Armando Machado (U.Lisboa)

Research articles in international journal and Preprints

28- Families of non-tiling domains satisfying Pólya's conjecture, (with P. Freitas). Preprint (2022), submited.
27- Pólya type inequalities on spheres and hemispheres, (with P. Freitas and J. Mao). Preprint (2021), submited.
26- Heinz mean curvature estimates in warped product spaces M × e f N . Annals of Global Analysis and Geometry 53 (2018), 265-281. 10.1007/s10455-017-9577-x
25- Dirichlet principal eigenvalue comparison theorems in geometry with torsion, (with A.C. Ferreira). Journal of Mathematical Analysis and Applications 453 2 (2017), :700-723. 10.1016/j.jmaa.2017.04.030.
24- Uniqueness of f-minimal hypersurfaces in warped product manifolds, (with J.J. Salamanca). Journal of Mathematical Analysis and Applications 422 2 (2015): 1376-1389. 10.1016/j.jmaa.2014.09.028
23- Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds, (with P. Freitas and J.Mao). Calculus of Variations and Partial Differential Equations, 51 (3-4)(2014): 701-724. 10.1007/s00526-013-0692-7
22- The isoperimetric problem in higher codimension (with F. Morgan). Manuscripta Mathematica 142(3-4) (2013): 369-382. 10.1007/s00229-012-0604-8
21- Stable 3-spheres in C3 . Journal of Mathematics Research 42 (2012):
http://www.ccsenet.org/journal/index.php/jmr/article/view/13403.
20- A spectral Bernstein theorem, (with P. Freitas). Annali di Matematica Pura ed Applicata 190 1 (2011), 77-90. 10.1007/s10231-010-0139-0
19- Mean curvature flow of spacelike graphs (with G. Li). Mathematische Zeitschrift 269 3-4 (2011): 697-719. 10.1007/s00209-010-0768-4
18- Erratum to: The G2 sphere of a 4-manifold , (with R. Albuquerque) Monatshefte Math., (2009), 158, (335-348), 10.1007/s00605-008-0053-3). Monat. fur Mathematik 160 1 (2010): 109-110. 10.1007/s00605-010-0191-2
17- Stability of submanifolds with parallel mean curvature in calibrated manifolds. Bulletin of the Brazilian Mathematical Society 41(4) (2010): 495-530. 10.1007/s00574-010-0023-y
16- Bernstein-Heinz-Chern results in calibrated manifolds, (with G. Li), Revista Matematica Iberoamericana 26(2) (2010). 651-692. https://projecteuclid.org/euclid.rmi/1275671315
15- Graphic Bernstein results in curved pseudo-Riemannian manifolds, (with G. Li). Journal of Geometry and Physics 59(9)(2009):1306-1313. 10.1016/j.geomphys.2009.06.011
14- The G2 sphere of a 4-manifold, (with R. Albuquerque). Monatshefte fur Mathematik 158 4 (2009): 335-348.   10.1007/s00605-008-0053-3
13- Forced convex mean curvature flow in Euclidean space, (with G.Li). Manuscripta Mathematica 126(3) (2008), 333-351. 10.1007/s00229-008-0181-z
12- Spacelike graphs with parallel mean curvature. Bulletin of the Belgian Mathematical Society - Simon Stevin 15(1) (2008), 65-76. https://projecteuclid.org/euclid.bbms/1203692447.
11- On the twistor space of pseudo-spheres, (with R. Albuqurque). Differential Geometry and its Application 25(2) (2007), 207-219. 10.1016/j.difgeo.2006.08.004
10- Space-time formulation for curvature flow of high codimension. (with G.Li) JP Journal of Geometry and Topology 6 no.3 (2006), p.303-318 6(3) (2006). 303-318. http://www.pphmj.com/journals/jpgt.htm.
9- Transgression forms in dimension 4, (with A. Pereira do Vale). International Journal of Geometric Methods in Modern Physics 3 (5-6) (2006). 1221-1254   10.1142/s0219887806001442
8- On the Kähler angles of submanifolds. Portugaliae Mathetmatica (N.S.) 60 (2) (2003),  215-235. https://www.emis.de/journals/PM/60f2/.
7- Minimal submanifolds of Kähler-Einstein manifolds with equal Kähler angles, (with G. Valli). Pacific Journal of Mathematics 205(1) (2002),  197-235. 10.2140/pjm.2002.205.197
6- Broadly-pluriminimal submanifolds of Kähler-Einstein manifolds, (with G. Valli). Yokohama Math. J. 48 (2001), no. 2, 181?199. 
5- On a class of graphs with prescribed mean curvature, (with Dong Minh Duc). Manuscripta Mathematica 82 1 (1994), 227-239.10.1007/BF02567699
4- Willmore submanifolds of the Möbius space and a Bernstein-type theorem, (with M. Rigoli) Manuscripta Mathematica 81(1)(1993): 203-222. 10.1007/BF02567854
3- Spherical-type hypersurfaces in a Riemannian manifold  (with J.-P. Ezin, and M. Rigoli); Israel Journal of Mathematics 71 (2) (1990): 193-209. 10.1007/BF02811884
2- Graphs with Parallel Mean Curvature, Proceedings of the American Mathematical Society 107 (2) (1989), 449-449. 10.2307/2047835
1- Conformal and isometric immersions of Riemannian manifolds, (with M. Rigoli). Mathematische Zeitschrift 196(2) (1987), 293-300. 10.1007/BF01163662

Thesis

Communications published in proceedings

4- Mean Curvature Flow and Bernstein-Calabi Results for Spacelike Graphs, (with G. Li),. VIII International Colloquium on Differential Geometry, Santiago de Compostela, Spain, 7-11 July 2008, satellite event of the 5th European Congress of Mathematics, Lopez, Jesus A. Alvarez(Ed), Garcia-Rio, Eduardo(Ed), World Scientific Publishing Company, April 2009, pp. 5164-174, arxiv:0810.3371
3 - Residue Formulas for Minimal Submanifolds, (with A. Pereira do Vale).  Proc. XV Intern. Workshop on Geometry and Physics, Tenerife, Spain, Sept. 11-16, 2006. Publ. de la RSME 11 (2007), 353-358.
2- Minimal submanifolds of Kähler- Einstein manifolds without complex directions and of half dimension, (with G. Valli). Summer School on Differential Geometry (Coimbra, 1999). ed. A.M. d'Ezevedo Breda, F.J. Craveiro de Carvalho, J.M. Nunes da Costa, Bernd Wegner, Coimbra, 3-7 September 1999. http://www.emis.de/proceedings/Coimbra99/index.html
1- Grassmannian manifolds as subsets of Euclidean spaces, (with A.Machado).Proc. 5th Int. Colloq., Santiago de Compostela, Spain, 1984, Research Notes in Mathematics, 131, 85-102, Pitman Adv. Pub. Prog., Boston-London-Melbourne (1985) https://arxiv.org/abs/2101.09731 http://www.ams.org/mathscinet-getitem?mr=864860.

Selected Invited Talks (not published)

5- Cheng's type comparison results for the principal eigenvalue of a V -drift Laplacian on a Riemannian manifold (with Ana C. Ferreira). International Meeting AMS/EMS/SPM, 10-13, June, 2015, Porto, Special Session 22 (Geometric Analysis) p. 39. Program abstract
4- Submanifolds with parallel mean curvature in calibrated manifolds: A variational approach, Third Minho Meeting on Mathematica Physics, Universidade do Minho, Braga, November 4, 2011.
3- The Spectrum of Minimal Submanifolds, invited talk at Dep. Geometria e Topologia, Universidad de Granada, Spain, 3 April 2009.
2- Angles of Minimal Submanifolds, invited talk at Workshop: Submanifolds Sendai 2004, Tohoku University, Sendai Japan, February 17-18, 2004. (*)
1- Minimal Submanifolds of Kähler-Einstein Manifolds, invited talk at Special Session on Ricci Curvature and Related Topics, 966th AMS Meeting, Stevens Institute of Technology, Hoboken, NJ, USA, April 28-29, 2001.

(*) with Prof. Katsuei Kenmotsu, Tohoku University, 2004

Supervision of Masters degree

M.Sc of Ana Mafalda Vaz de Araújo (Minimal Surfaces, Dec. 1990) Universidade de Lisboa.

M.Sc Rui Pedro L.P. Ribeiro de Albuquerque (Curvature of Left invariant Semi-Riemannian Lie groupes, Dec. 1996), Universidade de Lisboa

Supervision of PhD degree,  posgraduate and  post-doc

Posdoc researcher Guanghan Li (2005-2006) (IST fellowship) and (2007- 2008) Instituto Superior Técnico. (FCT fellowship SFRH/BPD/26554/2006).

(with Pedro Freitas) Jing Mao, who had a scholarship from FCT (SFRH/BD/60313/2009), as Ph.D. student in Mathematics at Instituto Superior Técnico, Lisbon (Eigenvalue Estimation and Some Results on Finite Topological Type) (2010-2013).

Juan J. Salamanca, Ph.D. student of Prof. Alfonso Romero, University of Granada,  IST (3 months) in 2014.

(with P. Freitas) posdoc Jing Mao,  Instituto Superior Técnico, 2018 with a grant from the Chinese Scholarship Council.

Member of exam committee

Member of the Ph.D. exam committee of Alma Luisa Albujer Brotons, Geometria global de superficies espaciales en espacios producto lorentziano,  Faculdad de Matematicas, Universidad Murcia, Spain (November 2008, student of Prof. L. Alias).

Member of the Ph.D. exam committee of Nuno Miguel Ferreira Correia, Aplicações Harmónicas de Superfícies de Riemann sobre Espaços Simétricos Universidade da Beira Interior, Covilhã (2012, student of Prof. R. Pacheco).

External Referee Report of Ph.D. thesis of Daniel de la Fuente Benito, Some problems on prescribed mean curvature and Kinematics in General Relativity, Universidad de Granada, Spain (2016, student of Prof. A. Romero).