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# Giona Veronelli

INdAM-COFUND Outgoing fellow between 2012-01-10 and 2012-08-31

### Project information

#### Title

SIRM: Sobolev Inequalities and Rigidity of Riemannian Manifolds

#### Outgoing host organisation Return host organisation

Université de Cergy-Pontoise Università degli Studi dell’Insubria

Boulevard du Port 33 Via Ravasi 2

95011 Cergy-Pontoise Cedex, France 21100 Varese, Italy

http://www.n.u-cergy.fr http://www.uninsubria.eu

#### Abstract

Sobolev inequalities are a fundamental tool which permits to link quantitatively Sobolev spaces with different orders of derivatives and integrability classes. They enable to improve analysison Riemannian manifolds supporting one of them. Hence, two natural problems require a thorough investigation, that is which manifolds support a Sobolev inequality and how its validity can be used to study geometric problems. This project aims to contribute on this topics following two main directions.

First, we would like to extend some existing rigidity results for manifolds supporting a Sobolev inequality with a specified value of the constants. In particular, we face two natural problems which are apparently unsolved, that is the case of a Euclidean Sobolev inequality on bounded domains of nonnegatively Ricci curved manifolds and the case of manifolds with Ricci curvature lower bounded by a negative constant and supporting a hyperbolic Sobolev inequality.

Then, we would like to use the validity of a Sobolev inequality on a given manifold to deduce new vanishing results for geometric quantities which are solutions of Bochner or Simons-type inequalities (e.g. harmonic or p-harmonic maps, second fundamental form, etc.). In particular we will focus on the likely nonlinear aspects of the techniques used so far in order to obtain a more suitable adaptation to p-harmonic realm. Finally, we will pursue the investigation of the analytic content of a celebrated vanishing theorem for minimal immersions in the attempt to get a deeper abstract comprehension of its geometric content and to obtain further geometric applications.

### Fellow information

#### Research interests

Main scientific interest in geometric analysis. In particular:

- Sobolev inequalities on manifolds
- Linear and semi-linear elliptic inequalities
- p-harmonic maps and p-Laplace operator
- f-harmonic maps and Ricci solitons
- Myers’ type theorems
- Convex bodies

#### Previous positions, awards

*January 2006*: Laurea triennale (bachelor) cum laude in Mathematics, Università dell’Insubria (Como, Italy)*October 2007*: Laurea specialistica (master degree) cum laude in Mathematics, Università dell’Insubria (Como, Italy)*February 2011*: Ph.D. in Mathematics, Università degli Studi di Milano, Italy. Thesis-title: Some analytic and geometric aspects of the p-Laplacian on Riemannian manifolds, Supervisor: Stefano Pigola*October 2010 – September 2011*: Post-Doc position under the supervision of Prof Emmanuel Hebey, AGM, Université de Cergy-Pontoise, France*October 2011- December 2011*: demi-ATER, AGM, Université de Cergy-Pontoise, France

#### Publications, preprints, other works

- S. Pigola, G. Veronelli,
*Uniform decay estimates for finite-energy*, Differential Geometry and Its Applications Journal,

solutions of semi-linear elliptic inequalities and geometric

applications

29 (2011), p. 35-54; DOI:10.1016/j.difgeo.2011.01.002 - S. Pigola, G. Veronelli,
*On the homotopy class of maps with finite p-energy*, Geometriae Dedicata 143 (2009), Issue 1,

into non-positively curved manifolds

p. 109-116, ISSN: 0046-5755; DOI:10.1007/s10711-009-9376-z - G. Veronelli,
*On p-harmonic maps and convex functions*, Manuscripta Math. 131 (2010), no. 3-4,

p. 537-546, ISSN: 0025-2611; DOI:10.1007/s00229-010-0335-7 - I. Holopainen, S. Pigola, G. Veronelli,
*Global comparison principles for the p-Laplace*, Potential Analysis 34 no. 4 (2011), p. 371-384; DOI:10.1007/s11118-010-9199-4

operator on Riemannian manifolds - G. Veronelli,
*Uniform decay estimates for solutions of the Yamabe equation*, Geometriae Dedicata 155 no. 1 (2011) 1-20; DOI:10.1016/j.difgeo.2011.01.002 - S. Pigola, G. Veronelli,
*Lower volume estimates and Sobolev inequalities*, Proc. Amer. Math. Soc. 138 (2010), p. 4479-4486; DOI:10.1090/S0002-9939-2010-10514-2 - P. Mastrolia, M. Rimoldi, G. Veronelli,
*Myers’ type theorems and some related oscillation results*, to appear in Journal of Geometric Analysis; DOI:10.1007/s12220-011-9213-0 - D. Valtorta, G. Veronelli,
*Stokes’ theorem, volume growth and parabolicity*, Tohoku Mathematical Journal, 63 no. 3 (2011), p. 397-412 - G. Veronelli,
*A global comparison theorem for p-harmonic maps in homotopy class*, Journal of Mathematical Analysis and Applications, 391 (2012) 335-349; DOI:10.1016/j.jmaa.2011.03.037 - S. Pigola, G. Veronelli,
*Remarks on L*, International Journal of Mathematics 23 no. 1 (2012) 1250008^{p}-vanishing results

in geometric analysis - E. Hebey, G. Veronelli,
*The Lichnerowicz equation in the closed case of the Einstein-Maxwell Theory*, accepted for publication in Transactions of the Amer. Math. Soc. - M. Rimoldi, G. Veronelli,
*f-harmonic maps and applications to gradient Ricci solitons*, preprint (2011); arXiv:1112.3637v1, submitted - S. Pigola, G. Veronelli,
*On the homotopy Dirichlet problem for p-harmonic maps*, preprint (2012); arXiv:1204.5430v1

#### Conferences, schools, other events

*ICTP-ESF School and Conference on Geometric Analysis*

Trieste, Italy, 11-29 June 2012*Recent trends in geometric and nonlinear analysis*

Banff International Research Station, Banff, Canada, 5-10 August 2012*Trimester in conformal and Kähler geometry*

IHP, Paris, France, 10 September – 14 December 2012