27,902 research outputs found
Combinatorial Yamabe flow on hyperbolic surfaces with boundary
This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with
boundary. It is proved by applying a variational principle that the length of
boundary components is uniquely determined by the combinatorial conformal
factor. The combinatorial Yamabe flow is a gradient flow of a concave function.
The long time behavior of the flow and the geometric meaning is investigated.Comment: 10 page
A characterization of hyperbolic geometry among Hilbert geometry
In this paper we characterize hyperbolic geometry among Hilbert geometry by
the property that three medians of any hyperbolic triangle all pass through one
point.Comment: 5 pages, 4 figure
Extremum problems for eigenvalues of discrete Laplace operators
The discrete Laplace operator on a triangulated polyhedral surface is related
to geometric properties of the surface. This paper studies extremum problems
for eigenvalues of the discrete Laplace operators. Among all triangles, an
equilateral triangle has the maximal first positive eigenvalue. Among all
cyclic quadrilateral, a square has the maximal first positive eigenvalue. Among
all cyclic -gons, a regular one has the minimal value of the sum of all
nontrivial eigenvalues and the minimal value of the product of all nontrivial
eigenvalues.Comment: 12 page
Geometric angle structures on triangulated surfaces
In this paper we characterize a function defined on the set of edges of a
triangulated surface such that there is a spherical angle structure having the
function as the edge invariant (or Delaunay invariant).
We also characterize a function such that there is a hyperbolic angle
structure having the function as the edge invariant.Comment: 10 page
A note on circle patterns on surfaces
In this paper we give two different proofs of Bobenko and Springborn's
theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle
pattern with proscribed intersection angles and cone angles on a cellular
decomposed surface up to isometry (or similarity).Comment: 16 pages, 2 figure
On parameterizations of Teichm\"uller spaces of surfaces with boundary
In \cite{rigidity}, Luo introduced a edge invariant which
turns out to be a coordinate of the Teichm\"uller space of a surface with
boundary. And he proved that for , the image of the
Teichm\"uller space under edge invariant coordinate is an open
cell. In this paper we verify his conjecture that for , the image of
the Teichm\"uller space is a bounded convex polytope.Comment: 12 pages, 4 figure
Canonical equations of Hamilton for the nonlinear Schr\"{o}dinger equation
We define two different systems of mathematical physics: the second-order
differential system (SODS) and the first-order differential system (FODS). The
Newton's second law of motion and the nonlinear Schr\"{o}dinger equation (NLSE)
are the exemplary SODS and FODS, respectively. We obtain a new kind of
canonical equations of Hamilton (CEH), which are of some kind of symmetry in
form and are formally different with the conventional CEH without symmetry [H.
Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison-Wesley,
2001]. We also prove that the number of the CEHs is equal to the number of the
generalized coordinates for the FODS, but twice the number of the generalized
coordinates for the SODS. We show that the FODS can only be expressed by the
new CEH, but do not by the conventional CEH, while the SODS can be done by both
the new and the conventional CEHs. As an example, we prove that the nonlinear
Schr\"{o}dinger equation can be expressed with the new CEH in a consistent way.Comment: 12 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1212.195
Cyclic polygons in classical geometry
Formulas about the side lengths, diagonal lengths or radius of the
circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or
spherical geometry can be unified.Comment: 8 pages, 3 figure
On a Conjecture of Milnor about Volume of Simplexes
We establish the second part of Milnor's conjecture on the volume of
simplexes in hyperbolic and spherical spaces. A characterization of the closure
of the space of the angle Gram matrices of simplexes is also obtained.Comment: 15 pages. Version for publicatio
Cell decompositions of Teichm\"uller spaces of surfaces with boundary
A family of coordinates for the Teichm\"uller space of a compact
surface with boundary was introduced in \cite{l2}. In the work \cite{m1},
Mondello showed that the coordinate can be used to produce a natural
cell decomposition of the Teichm\"uller space invariant under the action of the
mapping class group. In this paper, we show that the similar result also works
for all other coordinate for any .Comment: 11 pages, 6 figure
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