Let be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on is given.
We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on , where is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.
We define equivariant tensors for every non-negative integer and every Weil algebra and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type on an -dimensional manifold to tensor fields of type on if . Moreover, we determine explicitly the equivariant tensors for the Weil algebras , where and are non-negative integers.
We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Δh + 2h = 0 describing locally germs of minimal surfaces. Here Δ is the Laplace-Beltrami operator on the standard two-dimensional sphere. It explains the existence of the sum operator of minimal surfaces, introduced recently. In 4-dimensional space the equation Δ h + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero.
In the first part of this paper, we prove local interior and boundary gradient estimates for -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the -harmonic...
We give a sufficient condition for a curve to ensure that the -dimensional Hausdorff measure restricted to is locally monotone.
We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve , , is locally 1-monotone.