93 research outputs found

### The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian 3-dimensional
Heisenberg spaces ${\cal H}={\cal H}(\tau)$. Each such ${\cal H}$ is the total
space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with
geodesic fibers the orbits of a Killing field. We prove the existence and
uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian
submersion over certain domains $\Omega\subset\mathbb{R}^2$ taking on
prescribed boundary values

### Representations and classification of traveling wave solutions to Sinh-G{\"o}rdon equation

Two concepts named atom solution and combinatory solution are defined. The
classification of all single traveling wave atom solutions to Sinh-G{\"o}rdon
equation is obtained, and qualitative properties of solutions are discussed. In
particular, we point out that some qualitative properties derived intuitively
from dynamic system method aren't true. In final, we prove that our solutions
to Sinh-G{\"o}rdon equation include all solutions obtained in the paper[Fu Z T
et al, Commu. in Theor. Phys.(Beijing) 2006 45 55]. Through an example, we show
how to give some new identities on Jacobian elliptic functions.Comment: 12 pages. accepted by Communications in theoretical physics (Beijing

### A compactness theorem for complete Ricci shrinkers

We prove precompactness in an orbifold Cheeger-Gromov sense of complete
gradient Ricci shrinkers with a lower bound on their entropy and a local
integral Riemann bound. We do not need any pointwise curvature assumptions,
volume or diameter bounds. In dimension four, under a technical assumption, we
can replace the local integral Riemann bound by an upper bound for the Euler
characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.Comment: 28 pages, final version, to appear in GAF

### Parabolic stable surfaces with constant mean curvature

We prove that if u is a bounded smooth function in the kernel of a
nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian
manifold M, then u is either identically zero or it has no zeros on M, and the
linear space of such functions is 1-dimensional. We obtain consequences for
orientable, complete stable surfaces with constant mean curvature
$H\in\mathbb{R}$ in homogeneous spaces $\mathbb{E}(\kappa,\tau)$ with four
dimensional isometry group. For instance, if M is an orientable, parabolic,
complete immersed surface with constant mean curvature H in
$\mathbb{H}^2\times\mathbb{R}$, then $|H|\leq 1/2$ and if equality holds, then
M is either an entire graph or a vertical horocylinder.Comment: 15 pages, 1 figure. Minor changes have been incorporated (exchange
finite capacity by parabolicity, and simplify the proof of Theorem 1)

### Helicoidal surfaces rotating/translating under the mean curvature flow

We describe all possible self-similar motions of immersed hypersurfaces in
Euclidean space under the mean curvature flow and derive the corresponding
hypersurface equations. Then we present a new two-parameter family of immersed
helicoidal surfaces that rotate/translate with constant velocity under the
flow. We look at their limiting behaviour as the pitch of the helicoidal motion
goes to 0 and compare it with the limiting behaviour of the classical
helicoidal minimal surfaces. Finally, we give a classification of the immersed
cylinders in the family of constant mean curvature helicoidal surfaces.Comment: 21 pages, 22 figures, final versio

### The Abresch-Gromoll inequality in a non-smooth setting

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian CD(K,N) spaces in the same form as the one available on smooth Riemannian manifolds

### Asymptotically Extrinsic Tamed Submanifolds

We study, from the extrinsic point of view, the structure at infinity of open
submanifolds, ϕ : Mm → Mn(κ) isometrically immersed in the real space forms of
constant sectional curvature κ ≤ 0.We shall use the decay of the second fundamental
form of the so-called tamed immersions to obtain a description at infinity of the
submanifold in the line of the structural results in Greene et al. (Int Math Res Not
1994:364–377, 1994) and Petrunin and Tuschmann (Math Ann 321:775–788, 2001)
and an estimation from below of the number of its ends in terms of the volume growth
of a special class of extrinsic domains, the extrinsic balls.Vicent Gimeno: Work partially supported by the Research Program of University Jaume I Project UJI-B2016-07, and DGI -MINECO Grant (FEDER) MTM2013-48371-C2-2-P. Vicente Palmer: Work partially supported by the Research Program of University Jaume I Project UJI-B2016-07, DGI -MINECO Grant (FEDER) MTM2013-48371-C2-2-P, and Generalitat Valenciana Grant PrometeoII/2014/064. G. Pacelli Bessa: Work partially supported by CNPq- Brazil grant # 301581/2013-4

### Dirichlet sigma models and mean curvature flow

The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure

### Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

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