1. CMB 2015 (vol 58 pp. 723)
 Castro, Alfonso; Fischer, Emily

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear LaplaceBeltrami Equations on Spheres
We show that a class of semilinear LaplaceBeltrami equations
on the unit sphere
in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions.
The solutions to
these equations are the solutions to a two point boundary value
problem for a
singular ordinary differential equation. We prove the existence
of such solutions
using energy and phase plane analysis. We derive a
Pohozaevtype
identity
in
order to prove that the energy to an associated initial value
problem tends
to infinity as the energy at the singularity tends to infinity.
The nonlinearity is allowed to grow as fast as $s^{p1}s$ for
$s$ large
with $1 \lt p \lt (n+5)/(n3)$.
Keywords:LaplaceBeltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, subsuper critical nonlinearity Categories:58J05, 35A24 

2. CMB 2015 (vol 58 pp. 285)
 Karpukhin, Mikhail

Spectral Properties of a Family of Minimal Tori of Revolution in Fivedimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the LaplaceBeltrami
operator can be considered as functionals on the space of all
Riemannian metrics $g$ on a fixed surface $M$. In recent papers
several explicit examples of extremal metrics were provided.
These metrics are induced by minimal immersions of surfaces in
$\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family
of extremal metrics induced by minimal immersions in $\mathbb{S}^5$
is investigated.
Keywords:extremal metric, minimal surface Category:58J50 

3. CMB 2013 (vol 57 pp. 357)
 Lauret, Emilio A.

Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds
Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the
full isometry group $G$ of $\mathbb{R}^n$.
We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and
$\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups
$\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the
right regular representations $L^2(\Gamma_1\backslash G)$ and
$L^2(\Gamma_2\backslash G)$ are unitarily equivalent.
Keywords:representation equivalent, strongly isospectrality, compact flat manifolds Categories:58J53, 22D10 

4. CMB 2011 (vol 54 pp. 249)
5. CMB 2010 (vol 53 pp. 674)
 Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 

6. CMB 2010 (vol 53 pp. 684)
 Proctor, Emily; Stanhope, Elizabeth

An Isospectral Deformation on an InfranilOrbifold
We construct a Laplace isospectral deformation of metrics on an
orbifold quotient of a nilmanifold. Each orbifold in the deformation
contains singular points with order two isotropy. Isospectrality is
obtained by modifying a generalization of Sunada's theorem due to
DeTurck and Gordon.
Keywords:spectral geometry, global Riemannian geometry, orbifold, nilmanifold Categories:58J53, 53C20 

7. CMB 2009 (vol 52 pp. 66)
 Dryden, Emily B.; Strohmaier, Alexander

Huber's Theorem for Hyperbolic Orbisurfaces
We show that for compact orientable hyperbolic orbisurfaces, the
Laplace spectrum determines the length spectrum as well as the
number of singular points of a given order. The converse also holds, giving
a full generalization of Huber's theorem to the setting of
compact orientable hyperbolic orbisurfaces.
Keywords:Huber's theorem, length spectrum, isospectral, orbisurfaces Categories:58J53, 11F72 

8. CMB 2008 (vol 51 pp. 467)
 Wang, Yue

Coupled Vortex Equations on Complete KÃ¤hler Manifolds
In this paper, we first investigate the Dirichlet
problem for coupled vortex equations. Secondly, we give existence
results for solutions of the coupled vortex equations on a class
of complete noncompact K\"ahler manifolds which include
simplyconnected strictly negative curved manifolds, Hermitian
symmetric spaces of noncompact type and strictly pseudoconvex
domains equipped with the Bergmann metric.
Categories:58J05, 53C07 

9. CMB 2008 (vol 51 pp. 249)
 Mangoubi, Dan

On the Inner Radius of a Nodal Domain
Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 

10. CMB 2008 (vol 51 pp. 100)
 Petkov, Vesselin

Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to
several strictly convex disjoint obstacles is similar to that of the
inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function
$\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing
only over primitive periodic rays. In this paper we examine the
analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re
s = s_2$, where $s_2$ is the abscissa of absolute convergence of the
series obtained by the second iterations of the primitive periodic
rays. We show that at least one of the functions $Z_D(s), \Pi(s)$
has a singularity at $s = s_2$.
Keywords:dynamical zeta function, periodic rays Categories:11M36, 58J50 

11. CMB 2006 (vol 49 pp. 226)
 Engman, Martin

The Spectrum and Isometric Embeddings of Surfaces of Revolution
A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 

12. CMB 2002 (vol 45 pp. 378)
 FernándezLópez, Manuel; GarcíaRío, Eduardo; Kupeli, Demir N.

The Local MÃ¶bius Equation and Decomposition Theorems in Riemannian Geometry
A partial differential equation, the local M\"obius equation, is
introduced in Riemannian geometry which completely characterizes the
local twisted product structure of a Riemannian manifold. Also the
characterizations of warped product and product structures of
Riemannian manifolds are made by the local M\"obius equation and an
additional partial differential equation.
Keywords:submersion, MÃ¶bius equation, twisted product, warped product, product Riemannian manifolds Categories:53C12, 58J99 
