4,613 research outputs found
A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov
We prove a uniqueness theorem for immersed spheres of prescribed
(non-constant) mean curvature in homogeneous three-manifolds. In particular,
this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed
spheres of prescribed Weingarten curvature in R3 for the special but important
case of prescribed mean curvature. As a consequence, we extend the classical
Hopf uniqueness theorem for constant mean curvature spheres to the case of
immersed spheres of prescribed antipodally symmetric mean curvature in R3.Comment: 14 page
The Cauchy problem for Liouville equation and Bryant surfaces
We give a construction that connects the Cauchy problem for Liouville
elliptic equation with a certain initial value problem for mean curvature one
surfaces in hyperbolic 3-space H3, and solve both of them. We construct the
only mean curvature one surface in H3 that passes through a given curve with
given unit normal along it, and provide diverse applications. In particular,
topics like period problems, symmetries, finite total curvature, planar
geodesics, rigidity, etc. of surfaces are treated.Comment: 34 pages, 4 figure
Isometric immersions of R^2 into R^4 and pertubation of Hopf tori
We produce a new general family of flat tori in R^4, the first one since
Bianchi's classical works in the 19th century. To construct these flat tori,
obtained via small perturbation of certain Hopf tori in S^3, we first present a
global description of all isometric immersions of R^2 into R^4 with flat normal
bundle.Comment: 26 pages, 1 figur
Uniqueness of immersed spheres in three-manifolds
Let be a class of immersed surfaces in a three-manifold ,
and assume that is modeled by an elliptic PDE over each tangent
plane. In this paper we solve the so-called Hopf uniqueness problem for the
class under the only mild assumption of the existence of a
transitive family of candidate surfaces .
Specifically, we prove that any compact immersed surface of genus zero in the
class is a candidate sphere. This theorem unifies and extends
many previous uniqueness results of different contexts. As an application, we
settle in the affirmative a 1956 conjecture by A.D. Alexandrov on the
uniqueness of immersed spheres with prescribed curvatures in .Comment: 13 pages, 1 figur
Rotational symmetry of Weingarten spheres in homogeneous three-manifolds
Let be a simply connected homogeneous three-manifold with isometry group
of dimension , and let be any compact surface of genus zero
immersed in whose mean, extrinsic and Gauss curvatures satisfy a smooth
elliptic relation . In this paper we prove that is a
sphere of revolution, provided that the unique inextendible rotational surface
in that satisfies this equation and touches its rotation axis
orthogonally has bounded second fundamental form. In particular, we prove that:
(i) any elliptic Weingarten sphere immersed in
is a rotational sphere. (ii) Any sphere of constant positive extrinsic
curvature immersed in is a rotational sphere, and (iii) Any immersed sphere
in that satisfies an elliptic Weingarten equation
with bounded, is a rotational sphere. As a very particular case of this
last result, we recover the Abresch-Rosenberg classification of constant mean
curvature spheres in .Comment: 43 page
Serrin's overdetermined problem for fully nonlinear non-elliptic equations
Let denote a solution to a rotationally invariant Hessian equation
on a bounded simply connected domain , with
constant Dirichlet and Neumann data on . In this paper we
prove that if is real analytic and not identically zero, then is radial
and is a disk. The fully nonlinear operator is of
general type, and in particular, not assumed to be elliptic. We also show that
the result is sharp, in the sense that it is not true if is not simply
connected, or if is but not real analytic
Some Canonical Sequences of Integers
Extending earlier work of R. Donaghey and P. J. Cameron, we investigate some
canonical "eigen-sequences" associated with transformations of integer
sequences. Several known sequences appear in a new setting: for instance the
sequences (such as 1, 3, 11, 49, 257, 1531, ...) studied by T. Tsuzuku, H. O.
Foulkes and A. Kerber in connection with multiply transitive groups are
eigen-sequences for the binomial transform. Many interesting new sequences also
arise, such as 1, 1, 2, 26, 152, 1144, ..., which shifts one place left when
transformed by the Stirling numbers of the second kind, and whose exponential
generating function satisfies A'(x) = A(e^x -1) + 1.Comment: 18 pages, 2 figures; Dedicated to Professor J. J. Seide
Marginally trapped surfaces in L4 and an extended Weierstrass-Bryant representation
We give a conformal representation in terms of meromorphic data for a certain
class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean
curvature vector is either lightlike or zero at each point. This representation
extends simultaneously the Weierstrass representation for minimal surfaces in
Euclidean 3-space and for maximal surfaces in the Lorentz-Minkowski 3-space,
and the Bryant representation for mean curvature one surfaces in the hyperbolic
3-space and in the de Sitter 3-space.Comment: 20 page
A classification of isolated singularities of elliptic Monge-Amp\`ere equations in dimension two
Let denote the space of solutions to an elliptic,
real analytic Monge-Amp\`ere equation
whose graphs have a non-removable isolated singularity at the origin. We prove
that is in one-to-one correspondence with , where is a suitable subset of the class of regular, real
analytic strictly convex Jordan curves in . We also describe the
asymptotic behavior of solutions of the Monge-Amp\`ere equation in the
-smooth case, and a general existence theorem for isolated singularities
of analytic solutions of the more general equation
The geometric Neumann problem for the Liouville equation
In this paper we classify the solutions to the geometric Neumann problem for
the Liouville equation in the upper half-plane or an upper half-disk, with the
energy condition given by finite area. As a result, we classify the conformal
Riemannian metrics of constant curvature and finite area on a half-plane that
have a finite number of boundary singularities, not assumed a priori to be
conical, and constant geodesic curvature along each boundary arc
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