• Curriculum Vitae
  • Research

C.A.’s research activity mainly focuses on two themes, both related to analytic and geometric properties of Kaehler manifolds. The first one deals with the existence problem for special submanifolds of Einstein spaces, in particular of complex submanifolds, minima of the volume functional and their connection.
(cfr. Lawson e Simons, 1975, Siu e Yau, 1980, Micallef 1984, Wolfson 1989).
In Compositio Math. 1998 C.A. proved a Bernstein type theorem for stable minimal surfaces in hyperkaehler manifolds. C.A. has also developed a new existence theory for periodic minimal surfaces in euclidean spaces (J. Alg. Geom 1999, Int. J. Math. 2000, J. London. Math. Soc. 1999) and of minimal n-submanifold of C^n (Comm. Pure Appl. Math. 2003). In lavori Ann. Sc. Norm. Sup. Pisa 2000 and GAFA 2000, the first examples of nonholomorphic stable minimal surfaces in Einstein manifolds of negative or zero scalar curvature were given.
With G. La Nave (Adv Math 2005) we have shown that the same phenomenon occurs in Einstein Fano manifolds of complex dimension 3, therefore showing the optimality of Siu, Yau and Wolfson’s results. In the same paper examples of area minimizing symplectic nonholomorphic minimal two spheres in complex surfaces of constant positive scalar curvature are given.
The second theme of research deals with the existence of of special metrics on compact complex manifolds, namely balanced, extremal or Einstein.
In Ann. Sc. Norm. Sup. Pisa 2003, in collaboration with G. Tian, C.A. has given solutions of complex Monge-Ampere equations related to the existence of geodesics of kaehlerian potentials. In J. Geom. Phys. 2003 the theory of aproximation of kaehler metrics with projectively induced metrics (cfr. Tian 1989, Zelditch 1998) has been applied to some problems of geometric quantization.
In Comm. Math. Phys. 2004 with A. Loi, we have extended some results of Donaldson about uniqueness of balanced metrics to the case when the Kahler manifold has a non discrete automorphism group (similar results have been proved at the same time by Mabuchi).
In collaboration with A.Ghigi e G.P. Pirola (J. Crelle) we have recently constructed new Fano varieties which admit Kahler-Einstein metrics of positive curvature, finding in particular the first complete families of nonhomogeneous Fano varieties with a KE metric.
With F. Pacard (Paris) we have recently developed a general existence theory for constant scalar curvature Kahler metrics on blow ups of Kahler manifolds.

Served as Referee for the following Journals:

Inventiones Math., GAFA, IMRN, Comm. Math. Helvetici, Comm. Anal and Geom., Geometria
Dedicata, Journ. Symplectic Geom., Complex Variables and Elliptic Equations.

Co-organizer of international workshops at CIRM (Levico) and Banff (Canada).



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