54,192 research outputs found
On surfaces with prescribed shape operator
The problem of immersing a simply connected surface with a prescribed shape
operator is discussed. From classical and more recent work, it is known that,
aside from some special degenerate cases, such as when the shape operator can
be realized by a surface with one family of principal curves being geodesic,
the space of such realizations is a convex set in an affine space of dimension
at most 3. The cases where this maximum dimension of realizability is achieved
have been classified and it is known that there are two such families of shape
operators, one depending essentially on three arbitrary functions of one
variable (called Type I in this article) and another depending essentially on
two arbitrary functions of one variable (called Type II in this article).
In this article, these classification results are rederived, with an emphasis
on explicit computability of the space of solutions. It is shown that, for
operators of either type, their realizations by immersions can be computed by
quadrature. Moreover, explicit normal forms for each can be computed by
quadrature together with, in the case of Type I, by solving a single linear
second order ODE in one variable. (Even this last step can be avoided in most
Type I cases.)
The space of realizations is discussed in each case, along with some of their
remarkable geometric properties. Several explicit examples are constructed
(mostly already in the literature) and used to illustrate various features of
the problem.Comment: 43 pages, latex2e with amsart, v2: typos corrected and some minor
improvements in arguments, minor remarks added. v3: important revision,
giving credit for earlier work by others of which the author had been
ignorant, minor typo correction
Duality methods for a class of quasilinear systems
Duality methods are used to generate explicit solutions to nonlinear Hodge
systems, demonstrate the well-posedness of boundary value problems, and reveal,
via the Hodge-B\"acklund transformation, underlying symmetries among
superficially different forms of the equations.Comment: 14 page
Anholonomic Soliton-Dilaton and Black Hole Solutions in General Relativity
A new method of construction of integral varieties of Einstein equations in
three dimensional (3D) and 4D gravity is presented whereby, under corresponding
redefinition of physical values with respect to anholonomic frames of reference
with associated nonlinear connections, the structure of gravity field equations
is substantially simplified. It is shown that there are 4D solutions of
Einstein equations which are constructed as nonlinear superpositions of soliton
solutions of 2D (pseudo) Euclidean sine-Gordon equations (or of Lorentzian
black holes in Jackiw-Teitelboim dilaton gravity). The Belinski-Zakharov-Meison
solitons for vacuum gravitational field equations are generalized to various
cases of two and three coordinate dependencies, local anisotropy and matter
sources. The general framework of this study is based on investigation of
anholonomic soliton-dilaton black hole structures in general relativity. We
prove that there are possible static and dynamical black hole, black torus and
disk/cylinder like solutions (of non-vacuum gravitational field equations) with
horizons being parametrized by hypersurface equations of rotation ellipsoid,
torus, cylinder and another type configurations. Solutions describing locally
anisotropic variants of the Schwarzschild-- Kerr (black hole), Weyl
(cylindrical symmetry) and Neugebauer--Meinel (disk) solutions with anisotropic
variable masses, distributions of matter and interaction constants are shown to
be contained in Einstein's gravity. It is demonstrated in which manner locally
anisotropic multi-soliton-- dilaton-black hole type solutions can be generated.Comment: revtex, twocolumns, 24 pages, version 3 with minor correction
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