616 research outputs found
Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem
In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [31] and prove a
sharp convergence theorem for the mean curvature flow of arbitrary codimension
in spheres which improves the convergence theorem of Baker [3] as well as the
differentiable sphere theorems of Gu-Xu-Zhao [17, 51, 53].Comment: 20 page
A New Version of Huisken's Convergence Theorem for Mean Curvature Flow in Spheres
We prove that if the initial hypersurface of the mean curvature flow in
spheres satisfies a sharp pinching condition, then the solution of the flow
converges to a round point or a totally geodesic sphere. Our result improves
the famous convergence theorem due to Huisken [9]. Moreover, we prove a
convergence theorem under the weakly pinching condition. In particular, we
obtain a classification theorem for weakly pinched hypersurfaces. It should be
emphasized that our pinching condition implies that the Ricci curvature of the
initial hypersurface is positive, but does not imply positivity of the
sectional curvature.Comment: 20 page
An Optimal Convergence Theorem for Mean Curvature Flow of Arbitrary Codimension in Hyperbolic Spaces
In this paper, we prove that if the initial submanifold of dimension
satisfies an optimal pinching condition, then the mean curvature flow
of arbitrary codimension in hyperbolic spaces converges to a round point in
finite time. In particular, we obtain the optimal differentiable sphere theorem
for submanifolds in hyperbolic spaces. It should be emphasized that our
pinching condition implies that the Ricci curvature of the initial submanifold
is positive, but does not imply positivity of the sectional curvature of .Comment: 24 page
Mean curvature flow of arbitrary codimension in complex projective spaces
In this paper, we investigate the mean curvature flow of submanifolds of
arbitrary codimension in . We prove that if the initial
submanifold satisfies a pinching condition, then the mean curvature flow
converges to a round point in finite time, or converges to a totally geodesic
submanifold as . Consequently, we obtain a new
differentiable sphere theorem for submanifolds in . Our
work improves the convergence theorem for mean curvature flow due to Pipoli and
Sinestrari {\cite{PiSi2015}}.Comment: 31 page
A new pinching theorem for complete self-shrinkers and its generalization
In this paper, we firstly verify that if is a complete self-shrinker with
polynomial volume growth in , and if the squared norm of the
second fundamental form of satisfies , then
and is a round sphere or a cylinder. More generally, let
be a complete -hypersurface with polynomial volume growth in
with . Then we prove that there exists an
positive constant , such that if and the squared
norm of the second fundamental form of satisfies
, then ,
and is a cylinder. Here
.Comment: 14 page
On Chern's conjecture for minimal hypersurfaces in spheres
Using a new estimate for the Peng-Terng invariant and the multiple-parameter
method, we verify a rigidity theorem on the stronger version of Chern
Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that
if is a compact minimal hypersurface in whose squared
length of the second fundamental form satisfies ,
then and is a Clifford torus.Comment: 15 page
A new gap for complete hypersurfaces with constant mean curvature in space forms
Let be an -dimensional closed hypersurface with constant mean
curvature and constant scalar curvature in an unit sphere. Denote by and
the mean curvature and the squared length of the second fundamental form
respectively. We prove that if , where and , then . Here for , and
for . Moreover, we obtain a gap theorem for complete hypersurfaces with
constant mean curvature and constant scalar curvature in space forms.Comment: 13 page
Robust Online Matrix Factorization for Dynamic Background Subtraction
We propose an effective online background subtraction method, which can be
robustly applied to practical videos that have variations in both foreground
and background. Different from previous methods which often model the
foreground as Gaussian or Laplacian distributions, we model the foreground for
each frame with a specific mixture of Gaussians (MoG) distribution, which is
updated online frame by frame. Particularly, our MoG model in each frame is
regularized by the learned foreground/background knowledge in previous frames.
This makes our online MoG model highly robust, stable and adaptive to practical
foreground and background variations. The proposed model can be formulated as a
concise probabilistic MAP model, which can be readily solved by EM algorithm.
We further embed an affine transformation operator into the proposed model,
which can be automatically adjusted to fit a wide range of video background
transformations and make the method more robust to camera movements. With using
the sub-sampling technique, the proposed method can be accelerated to execute
more than 250 frames per second on average, meeting the requirement of
real-time background subtraction for practical video processing tasks. The
superiority of the proposed method is substantiated by extensive experiments
implemented on synthetic and real videos, as compared with state-of-the-art
online and offline background subtraction methods.Comment: 14 pages, 13 figure
Toward Real-World Single Image Super-Resolution: A New Benchmark and A New Model
Most of the existing learning-based single image superresolution (SISR)
methods are trained and evaluated on simulated datasets, where the
low-resolution (LR) images are generated by applying a simple and uniform
degradation (i.e., bicubic downsampling) to their high-resolution (HR)
counterparts. However, the degradations in real-world LR images are far more
complicated. As a consequence, the SISR models trained on simulated data become
less effective when applied to practical scenarios. In this paper, we build a
real-world super-resolution (RealSR) dataset where paired LR-HR images on the
same scene are captured by adjusting the focal length of a digital camera. An
image registration algorithm is developed to progressively align the image
pairs at different resolutions. Considering that the degradation kernels are
naturally non-uniform in our dataset, we present a Laplacian pyramid based
kernel prediction network (LP-KPN), which efficiently learns per-pixel kernels
to recover the HR image. Our extensive experiments demonstrate that SISR models
trained on our RealSR dataset deliver better visual quality with sharper edges
and finer textures on real-world scenes than those trained on simulated
datasets. Though our RealSR dataset is built by using only two cameras (Canon
5D3 and Nikon D810), the trained model generalizes well to other camera devices
such as Sony a7II and mobile phones
Ancient Solution of Mean Curvature Flow in Space Forms
In this paper we investigate the rigidity of ancient solutions of the mean
curvature flow with arbitrary codimension in space forms. We first prove that
under certain sharp asymptotic pointwise curvature pinching condition the
ancient solution in a sphere is either a shrinking spherical cap or a totally
geodesic sphere. Then we show that under certain pointwise curvature pinching
condition the ancient solution in a hyperbolic space is a family of shrinking
spheres. We also obtain a rigidity result for ancient solutions in a
nonnegatively curved space form under an asymptotic integral curvature pinching
condition.Comment: 23 page
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