Showing papers in "Journal of Differential Geometry in 1971"

Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds

Abstract: Let (M,g) denote a smooth (say C) compact two-dimensional manifold, equipped with some Riemannian metric g. Then, as is well-known, M admits a metric gc of constant Gaussian curvature c in fact the metrics g and gc can be chosen to be conformally equivalent. Here, we determine sufficient conditions for a given non-simply connected manifold M to admit a Riemannian structure g (conformally equivalent to g) with arbitrarily prescribed (Holder continuous) Gaussian curvature K(x). If the Euler-Poincare characteristic χ(M) of M is negative, the sufficient condition we obtain is that K(x) < 0 over M. Note that this condition is independent of g, and this result is obtained by solving an isoperimetric variational problem for g. If K(x) is of variable sign for χ(M) < 0, or if χ(M) > 0, then the desired critical point may not be an absolute minimum and our methods do not succeed. If χ(M) = 0, our methods apply when K(x) satisfies an integral condition with respect to the given metric g (see § 3) this result is perhaps not unreasonable since, for χ(M) < 0, distinct Riemannian structures on M need not be conformally equivalent.

Submanifolds with parallel mean curvature vector

Abstract: J. Simons [5] has recently proved a formula which gives the Laplacian of the square of the length of the second fundamental form, and applied the formula to the study of minimal hypersurfaces in the sphere (see also [1], [2]). K. Nomizu and B. Smyth [4] have obtained a formula of the same type for a hypersurface immersed with constant mean curvature in a space of constant sectional curvature, and derived a new formula for the Laplacian of the square of the length of the second fundamental form, in which the sectional curvature of the hypersurface appears. Using this new formula, they determined hypersurfaces of nonnegative sectional curvature and constant mean curvature immersed in the Euclidean space or in the sphere under the additional condition that the square of the length of the second fundamental form is constant. The purpose of the present paper is to generalize Nomizu-Smyth formulas to the case of general submanifolds and to use the formulas to study submanifolds, immersed in a space of constant curvature, whose normal bundle is locally parallelizable and mean curvature vector field is parallel in the normal bundle.

A remark on Holmgren's uniqueness theorem

Abstract: In a recent note Bony [1] has given a remarkable improvement of Holmgren's uniqueness theorem. The result is as follows. Let P(x, D) be a differential operator with analytic coefficients in a neighborhood X of a point x0 e R , and denote the principal symbol by Pm(x, ξ) where xeX and ξ e R . Let u e &(X) be a solution of the equation P(x,D)u = 0 vanishing when φ(x0), xeX, where φ e C\\X) and A ^ g r a d φ(x0)φ0. Holmgren's uniqueness theorem then states that u must vanish in a neighborhood of xQ if Pm(x0, No) Φ 0. (Schapira [4] has proved that this remains true for hyperfunction solutions.) Bony [1] introduced the smallest ideal I(P) in C°°(X X (R\\0), R) such that (i) Q e I(P) if Q(x, ξ) is positively homogeneous with respect to ξ and vanishes for all (JC, f ) e J f χ (R\\0) with PTO(JC, f) = 0, (ϋ) β i , β 2

Rigidity of hypersurfaces of constant scalar curvature

Abstract: Let Mn and Mn+i be Riemannian manifolds of dimension n and n+1 respectively. Assume Mn isometrically immersed in Mn+iIf each point p of Mn is contained in an open neighborhood U This concept allowed us to show that the result of NaganoTakahashi [3] holds without any restriction, i.e. that any homogeneous hypersurface of the Euclidean space JSn+i is isometric to the Riemannian product of a ^-dimensional sphere and an n—p dimensional Euclidean space. This result is a consequence of the following