We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_κ= \big ( \sum_{i < j} \frac{1}{λ_i+λ_j-2κ} \big )^{-1}$, where $λ_1 \leq \hdots \leq λ_n$ denote the curvature eigenvalues and $κ$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $λ_1+λ_2>2κ$. In contrast to mean curvature flow, this flow preserves the condition $λ_1+λ_2>2κ$ in a general ambient manifold.
Our main goal in this paper is to extend the surgery algorithm of Huisken-Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $\overline{R}_{1313}+\overline{R}_{2323} \geq -2κ^2$ with non-empty boundary satisfying $λ_1+λ_2 > 2κ$ is diffeomorphic to a $1$-handlebody.
The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken-Sinestrari, Haslhofer-Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.
We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.
Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures
We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is $σ_k$-convex.
The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.
In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12].
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.
In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.
We consider the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the $H^α$-flow is established by means of approximation by bounded problems.
We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds
We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.
In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p
We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.
In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.
We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.
We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.
We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n \times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.
In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere \cite{gerh,Mak-Sch} to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in hyperbolic space and solve it for curvature functions $F$ the inverses of which are of class $(K^*)$.
For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.
A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermo…
We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $Σ$ immersed in $\mathcal{C}$ with free boundary in $\partial\mathcal{C}\setminus\{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape.
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannia…
In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfaces are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its topology.
We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $σ_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $\mathbb{R}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times\mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $σ_{n-1}$ curvature and bounded principal curvatures.
This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze \cite{Sch05} for the positive power mean curvature flow to a much…
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.
Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing this $G$-flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex s…
We prove that any $n$-dimensional closed mean convex $λ$-hypersurface is convex if $λ\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $λ$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $λ$-hypersurfaces with $λ\le 0.$
We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove th…
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for insta…
We show that for a very general and natural class of curvature functions (for example the curvature quotients $(σ_n/σ_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(κ) = σ\in (1,\infty)$ with a prescribed compact future asymptotic boundary $Γ$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly con…
Let $M^{2n+1}$, $n\ge 1$, be a smooth manifold with a pseudo-convex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $ M=\mathcal S_0 \supset \mathcal S_1 \supset \cdots \supset \mathcal S_{n}, $ where $\mathcal S_q$ is the set of points where the Levi-form has nullity $\ge q$. We prove that $\mathcal S_q$'s are locally given as common zero sets of the coefficients $A_j,$ $j=0,1,\ldots, q-1,$ of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coeffici…
In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.
Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.
We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism between the fundamental groups of S and N induced by the inclusion is surjective, provided only that N - S has two connected components, one of which has noncompact closure and trivial homotopy relative to S. The idea of the proof is to view N as a spacelike hypersurface in a suitable Lorentz manifold and then apply a recent version of certain classic results b…
