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Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second…
We study new examples of translating solitons of the mean curvature flow, especially in Minkowski space. We consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This…
In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all…
We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity…
We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are…
If $\xi$ is a Killing vector field of the hyperbolic space $\h^3$ whose flow are parabolic isometries, a surface $\Sigma\subset\h^3$ is a $\xi$-translator if its mean curvature $H$ satisfies $H=\langle N,\xi\rangle$, where $N$ is the unit…
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative…
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…
In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in…
In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap…
Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as "localised`` versions…
In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdzi\'nski [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$. More…
The aim of this paper is to present a complete description of all rotational linear Weingarten surface into the Euclidean sphere S3. These surfaces are characterized by a linear relation aH+bK=c, where H and K stand for their mean and…
We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit…
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…
We establish the existence of one-parameter families of helicoidal surfaces of $\mathbb H^2\times\mathbb R$ which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.
In this paper we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flows given by a homogeneous, symmetric, smooth non-negative function $\gamma$ defined in an open cone…
A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…
There exist four non-equivalent types of the translation hypersurfaces in the 4-dimensional isotropic space $\mathbb{I}^{4}$ generated by translating the curves lying in perpendicular $k-$planes $\left(k=2,3\right)$, due to its absolute…
We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the…