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Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 61 (10 self)
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are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the pmean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining
Minimal surfaces in pseudohermitian geometry
, 2004
"... We consider surfaces immersed in threedimensional pseudohermitian manifolds. We define the notion of (p)mean curvature and of the associated (p)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the pmean curvature not only as the tangential ..."
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We consider surfaces immersed in threedimensional pseudohermitian manifolds. We define the notion of (p)mean curvature and of the associated (p)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the pmean curvature not only as the tangential
DOI: 10.1007/s0052600302104
, 2003
"... Abstract. We give a new proof of regularity of biharmonic maps from fourdimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of sol ..."
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Abstract. We give a new proof of regularity of biharmonic maps from fourdimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives. Mathematics Subject Classification (2000): 35J60, 35H20 1.