1,720,998 research outputs found
Index bounds for minimal hypersurfaces of the sphere
No full textWe consider a compact, orientable minimal hypersurfaces of the unit sphere and prove a comparison theorem between the spectrum of the stability operator and that of the Laplacian on 1-forms. As a corollary, we show that the index is bounded below by a linear function of the first Betti number; in particular, if the first Betti number is large, then the immersion is highly unstable
On the first hodge eigenvalue of isometric immersions
No full textWe give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on p-forms on a compact manifold without boundary isometrically immersed in Rn or Sn. The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature
Heat flow, heat content and the isoparametric property
Full text linkLet M be a Riemannian manifold and Ω a compact domain of M with smooth boundary.
We study the solution of the heat equation on Ω having constant unit initial conditions
and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of
domains for which, at any fixed value of time, the normal derivative of the solution (heat
flow) is a constant function on the boundary. We express this fact by saying that such
domains have the
constant flow property
. In constant curvature spaces known examples of
such domains are given by geodesic balls and, more generally, by domains whose boundary
is connected and isoparametric. The question is: are they all like that? This problem is the
analogous (for the heat equation) of the classical Serrin’s problem for harmonic domains.
In this paper we give an affirmative answer to the above question: in fact we prove
more generally that, if a domain in an analytic Riemannian manifold has the constant flow
property, then every component of its boundary is an isoparametric hypersurface. For space
forms, we also relate the order of vanishing of the heat content with fixed boundary data with
the constancy of the r-mean curvatures of the boundary and with the isoparametric property.
Finally, we discuss the constant flow property in relation to other well-known overdetermined
problems involving the Laplace operator, like the Serrin problem or the Schiffer proble
Lower bounds for the nodal length of eigenfunctions of the Laplacian
Full text linkWe prove lower bounds for the length of the zero set of an eigenfunction of the Laplace operator on a Riemann surface; in particular, in non-negative curvature, or when the associated eigenvalue is large, we give a lower bound which involves only the square root of the eigenvalue and the area of the manifold (modulo a numerical constant, this lower bound is sharp)
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