By William Anthony Granville
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Additional info for Elements of the Differential and Integral Calculus
43 [Mill P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. , 130 (1995) 334-344. [Mi2] P. Mironescu, Les minimiseurs locaux pour l'fiquation de Ginzburg-Landau sont 5. symdtrie radiale, C. R. Acad. Sci. Paris, 6, (323), 593-598. [PN] L. Pismen and A. Nepomnyashechy, Stability of vortex rings in a model of superflow, Physica D, (1993) 163-171. [RS] J. Rubinstein and P. Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, to appear.
M". 1) The problem with the degeneracy ill tangential direction can be overcome in various ways: In  Hamilton solves degenerate parabolic equations satisfying an integrability condition. 1) holds. The other approach consists in the choice of some vectorfield transversal to tile initial surface. 1) holds. This approach was originally used by DeTurck  for the Ricci flow and has been used for the evolution of hypersurfaces in , . In the last chapter of the present paper the second author gives a selfcontained proof of shorttime existence for a large class of geometric evolution equations including equations of higher order.
For this purpose evolution equations have to be established for all relevant geometric quantities, in particular for the second fundamental form. I) the following equations hold: (i) -~g~i~ = 2fh~, (ii) a(d#) = fU(d#), (iii) a v =-V (iv) ~ f, = - V , V j f -t- f(h,kh~ - Rowj), (v) ~ I f = - A f - f(tml 2 + ~ c ( . )). llcrc dlz is the induced mcasure on the hypcrsurface and A is the Laplace-Beltrami operator with respect to the time-dependent induced metric on the hypersurface. Notice that - A f - f(lAI 2 + l~ic(~, ~)) = J f is the Jacobi operator acting on f , as is wellknown from the second variation formula for the area.
Elements of the Differential and Integral Calculus by William Anthony Granville