1,970 research outputs found
Vertex theorems for capillary drops on support planes
We consider a capillary drop that contacts several planar bounding walls so
as to produce singularities (vertices) in the boundary of its free surface. It
is shown under various conditions that when the number of vertices is less than
or equal to three, then the free surface must be a portion of a sphere. These
results extend the classical theorem of H. Hopf on constant mean curvature
immersions of the sphere. The conclusion of sphericity cannot be extended to
more than three vertices, as we show by examples
Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds
Given a domain of and a -dimensional
non-degenerate minimal submanifold of \pa \Omega with , we
prove the existence of a family of embedded constant mean curvature
hypersurfaces which as their mean curvature tends to infinity concentrate along
and intersecting perpendicularly.Comment: 28 page
Faraday instability on a sphere: numerical simulation
We consider a spherical variant of the Faraday problem, in which a spherical
drop is subjected to a time-periodic body force, as well as surface tension. We
use a full three-dimensional parallel front-tracking code to calculate the
interface motion of the parametrically forced oscillating viscous drop, as well
as the velocity field inside and outside the drop. Forcing frequencies are
chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We
excite gravity waves for wavenumbers 1 and 2 and observe translational and
oblate-prolate oscillation, respectively. For wavenumbers 3 to 6, we excite
capillary waves and observe patterns analogous to the Platonic solids. For low
viscosity, both subharmonic and harmonic responses are accessible. The patterns
arising in each case are interpreted in the context of the theory of pattern
formation with spherical symmetry
Translating solitons over Cartan-Hadamard manifolds
We prove existence results for entire graphical translators of the mean
curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We
show that the asymptotic behaviour of entire solitons depends heavily on the
curvature of the manifold, and that there exist also bounded solutions if the
curvature goes to minus infinity fast enough. Moreover, it is even possible to
solve the asymptotic Dirichlet problem under certain conditions.Comment: This replaces the first version. We have deleted the whole Section 3
from the previous version due to a gap in a proof. We are grateful to Dr.
Hengyu Zhou for pointing out the gap in the proof of Lemma 3.3 in the
previous versio
Equilibrium of Surfaces in a Vertical Force Field
Funding for open access charge: Universidad de Granada / CBUA.The authors are grateful to Margarita Arias, Jos´e Antonio G´alvez and Francisco Martín for helpful comments during the preparation of this manuscript.In this paper, we study phi-minimal surfaces in R-3 when the function phi is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in R-2. We describe a full classification of complete flat-embedded phi-minimal surfaces if phi is strictly monotone and characterize rotational phi-minimal surfaces by its behavior at infinity when phi has a quadratic growth.Universidad de Granada / CBU
Multi-scale spectral methods for bounded radially symmetric capillary surfaces
We consider radially symmetric capillary surfaces that are described by
bounded generating curves. We use the arc-length representation of the
differential equations for these surfaces to allow for vertical points and
inflection points along the generating curve. These considerations admit
capillary tubes, sessile drops, and fluids in annular tubes as well as other
examples.
We present a multi-scale pseudo-spectral method for approximating solutions
of the associated boundary value problems based on interpolation by Chebyshev
polynomials. The multi-scale approach is based on a domain decomposition with
adaptive refinements within each sub-domain.Comment: arXiv admin note: text overlap with arXiv:2205.0293
Scalar curvature comparison of rotationally symmetric sets
Let be a compact 3-manifold with nonnegative scalar curvature
. The boundary is diffeomorphic to the boundary of a
rotationally symmetric and weakly convex body in . We
call a model or a reference. Let and
be respectively the mean curvatures of in
and in , and
be the induced metric from and . We show that for some classes of
, if , and the dihedral angles at the nonsmooth part of are
no greater than the model, then is flat. We also generalize this result to
the hyperbolic case and some spaces with -symmetry. Our approach
is inspired by Gromov.Comment: 48pages, 4 figures. All comments are welcom
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