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Departamento de Matematática

Instituto de Matemática e Estatística

Universidade de São Paulo

Rua do Matão, 1010

São Paulo - SP - Brasil - CEP 05508-090

- Office 164 in the Bloco B of IME
- email: clarke at ime.usp.br

I am currently (until March 2013) a post-doctoral fellow at the Universidade de São Paulo (Brazil), invited by Professor Claudio Gorodski. I am a member of the geometry research group in the Departamento de Matemática within the Instituto de Matemática e Estatística.From April 2013 I am professor adjunto at the Universidade Federal do Rio de Janeiro.

The geometry of submanifolds of riemannian manifolds, isolation phenomena and the rigidity of submanifolds of symmetric spaces of compact type, the geometry of manifolds of reduced holonomy, twistor theory and minimal surfaces of manifolds of G2 holonomy, gauge theory on manifolds of G2 holonomy, certain constructions of instantons for special metrics.

- Rigidity of rank-one factors of compact symmetric spaces
*( Ann. de l'Inst. Fourier, Vol. 61 no. 2 (2011), p. 491-509)*We consider the decomposition of a riemannian symmetric space of compact type as a product of factors and show that, with the exception of the Cayley plane, the factors of rank one are rigid when considered as minimal submanifolds. The proof is initially done for the case that the factor is a sphere. Then we recall that the other spaces of rank one admit a Hopf fibration from a sphere to finish the proof in this case. - The Perron method and the non-linear Plateau problem
*(with Graham Smith, Geom. Ded., Vol. 163, no. 1 (2013), p 159-164)*We give a technique inspired by the recent work of Harvey and Lawson to resolve the plateau problem for hypersurfaces of constant curvature. This is illustrated by an example of of constant Gaussian curvature in*R^{n+1}*. - Minimal surfaces in G2 manifolds
*(submitted)*We define the twistor space*Z_X*of a manifold*X*of*G_2*holonomy and show that it admits a distribution of codimension one that supports a hermitian structure. For every Riemann surface*S*immersed in*X*there exists a lifting*S'*to*Z_X*that is tangent to the distribution. If*S*is an*adapted*surface in*X*, it is a*minimal*surface if and only if the lifting is a pseudo-holomorphic curve. - Lower bounds on the modified
*K*-energy and complex deformations*(submitted)*Let*(X,L)*be a polarized Kähler manifold that admits an extremal Kähler metric in*c_1(L)*. We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of*(X,L)*, the modified*K*-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti to extremal metrics. Our proof also extends a convexity inequality on the space of Kähler potentials due to X.X. Chen to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of*CP^1\times CP^1*that carry no extremal metric but with modified*K*-energy bounded from below.

*G_2*Instantons on the manifolds of Bryant and Salamon*(in preparation)*- Instantons on the Kummer Surface
*(in preparation)*