On deformations of closed surfaces of genus $p\geqslant1$ with given infinitesimal change of metric

In this article the author studies deformations with given infinitesimal change of metric of a closed surface of genus $p\geqslant1$ of positive extrinsic curvature, situated in a three-dimensional Riemannian space. It is established that, in contrast to the case $p=0$ (investigated by H. Weyl and A. V. Pogorelov), the surface does not admit deformations with an arbitrary preassigned infinitesimal change of metric. Conditions are obtained on the given change of metric that are necessary and sufficient for the existence of a deformation. As an auxiliary result necessary and sufficient conditions are established for the existence on a closed Riemann surface of a regular global solution of a nonhomogeneous elliptic system of Carleman type.
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