For two-dimensional, immersed closed surfaces $f:\Sigma \to {ℝ}^n$, we study the curvature functionals ${ℰ}^p\left(f\right)$ and ${𝒲}^p\left(f\right)$ with integrands ${(1+|A|}^2{)}^{p/2}$ and ${(1+|H|}^2{)}^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p>2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for ${𝒲}^p$-bounded sequences. In the case of ${ℰ}^p$ this is just Langer’s theorem [16], while for ${𝒲}^p$ we have to impose a bound for the Willmore energy strictly below $8\pi$ as an additional...

Given a domain $\Omega$ of ${ℝ}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\partial \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces in $\Omega$ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly along their boundaries.

We construct travelling wave graphs of the form $z=-ct+\phi \left(x\right)$, $\phi :x\in {ℝ}^{N-1}\mapsto \phi \left(x\right)\in ℝ$, $N\ge 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\ge c_0$) with prescribed asymptotics. For any $1$-homogeneous function ${\phi }_{\infty }$, viscosity solution to the eikonal equation $|D{\phi }_{\infty }|=\sqrt{{(c/c_0)}^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by ${\phi }_{\infty }$. We also describe ${\phi }_{\infty }$ in terms of a probability measure on ${\S }^{N-2}$.

A classical result of A. D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in ${ℝ}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=\text{const}$ in case $M$ satisfies: for any two points $(X^{\text{'}},X_{n+1})$, $(X^{\text{'}},{\widehat{X}}_{n+1})$ on $M$, with $X_{n+1}>{\widehat{X}}_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$....

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–{\Delta }_{p}u=g\left(u\right)$ in a smooth bounded domain $\Omega \subset {ℝ}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^{☆}$ when the domain is strictly convex. More precisely, we prove that $u^{☆}\in L^{\infty }\left(\Omega \right)$ if $n\le p+2$ and $u^{☆}\in L^{\frac{np}{n-p-2}}\left(\Omega \right)\cap W_0^{1,p}\left(\Omega \right)$ if $n>p+2$.

Let M be a closed connected surface in $R^3$ with positive Gaussian curvature K and let $K_{I}I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I}I=c\surd H{K}^r$, for some constants c and r, where H is the mean curvature of M.

Let Mⁿ be a hypersurface in $R^{n+1}$. We prove that two classical Jacobi curvature operators $J_x$ and $J_y$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $\left(K_{x,y}\circ K_{z,u}\right)\left(u\right)=\left(K_{z,u}\circ K_{x,y}\right)\left(u\right)$, where $K_{x,y}\left(u\right)=R(x,y,u)$, for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

Let $(M,g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$. In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $\mathrm{O}\left(4\right)$ acts as a transformation group between ST bases at $T_{p}M$ and for the so-called 2-stein curvature tensors, the group $\mathrm{Sp}\left(1\right)\subset \mathrm{SO}\left(4\right)$ acts as a transformation...

Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n+p}\left(1\right)$. By using the Sobolev inequalities of P. Li to get $L_p$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${\left|\right|\sigma \left|\right|}_p$ the mean curvature and the $L_p$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${\left|\right|\sigma \left|\right|}_{n/2}<C$, then M is a minimal submanifold in the sphere $S^{n+p-1}(1+H²)$ with sectional...

We extend a result of the second author [27, Theorem 1.1] to dimensions $d\ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<2(d+1)/d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

Let $E=\{u\in C^1[0,1]:u\left(0\right)=u\left(1\right)=0\}$. Let $S_k^{\nu }$ with $\nu =\{+,-\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^{\nu }$-solutions of the problem $$\left\{\begin{array}{cc}{\left({\displaystyle \frac{u^{\text{'}}}{\sqrt{1-u^{\text{'}2}}}}\right)}^{\text{'}}+\lambda a\left(x\right)f\left(u\right)=0,\hfill & x\in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0,\hfill \end{array}\right.$$ where $\lambda >0$ is a parameter, $a\in C\left(\right[0,1],(0,\infty \left)\right)$. We determine the intervals of parameter $\lambda$ in which the above problem has one, two or three $S_k^{\nu }$-solutions. The proofs of the main results are based upon the bifurcation technique.

Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k\left(t\right)$ with respect to some point $p\in M$, where $t=d(·,p)$ is the Riemannian distance on $M$ to $p$, $k\left(t\right)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty$, satisfy $$\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}+\cdots +\frac{1}{{\mu }_n}\ge \frac{l^{n+2}}{(n+2){\int }_0^l{f}^{n-1}\left(t\right)\mathrm{d}t},$$ where $f\left(t\right)$ is the solution to $$\left\{\begin{array}{c}{f}^{\text{'}\text{'}}\left(t\right)+k\left(t\right)f\left(t\right)=0\text{on}(0,\infty ),\hfill \\ f\left(0\right)=0,f^{\text{'}}\left(0\right)=1.\hfill \end{array}\right.$$

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset {ℝ}^2$. We prove that, for such $A$, the distance function $d_A=\mathrm{dist}(·,A)$ is a “DC aura” for $A$, which implies that each closed locally WDC set in ${ℝ}^2$ is a WDC set. Another consequence is that compact WDC subsets of ${ℝ}^2$ form a Borel subset of the space of all compact sets.

We study the notion of strong $r$-stability for the context of closed hypersurfaces ${\Sigma }^n$ ($n\ge 3$) with constant $(r+1)$-th mean curvature $H_{r+1}$ immersed into the Euclidean sphere ${𝕊}^{n+1}$, where $r\in \{1,...,n-2\}$. In this setting, under a suitable restriction on the $r$-th mean curvature $H_r$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of ${𝕊}^{n+1}$, a region that is determined by a totally umbilical sphere of ${𝕊}^{n+1}$. We also provide a rigidity result for such hypersurfaces.

Let $S$ be a Riemann surface. Let ${ℍ}^3$ be the $3$-dimensional hyperbolic space and let ${\partial }_{\infty }{ℍ}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial }_{\infty }{ℍ}^3=\widehat{ℂ}$. If $i:S\to {ℍ}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{ı}$, of $i$ by $\widehat{ı}=N$. Let $\overrightarrow{n}:U{ℍ}^3\to {\partial }_{\infty }{ℍ}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{ı})$ is complete and $\overrightarrow{n}\circ \widehat{ı}=\varphi$. In this...

We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\overline{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $\zeta$ of $\overline{M}$ is tangent to $M$. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\overline{M}$ such that (1) the screen distribution $S\left(TM\right)$ is totally umbilical or (2) $M$ is screen conformal.

The numerical approximation of the minimum problem: ${min}_{A\subseteq \mathrm{\Omega }}\tilde{\mathcal{F}}\left(A\right)$, is considered, where $\tilde{\mathcal{F}}\left(A\right)=P_{\mathrm{\Omega }}\left(A\right)+\cos \left(\theta \right){\mathcal{H}}^{n-1}\left(\partial A\cap \partial \mathrm{\Omega }\right)-{\int }_A\kappa$. The solution to this problem is a set $A\subseteq \mathrm{\Omega }\subset {\mathbb{R}}^n$ with prescribed mean curvature $\kappa$ and contact angle $\theta$ at the intersection of $\partial A$ with $\partial \mathrm{\Omega }$. The functional $\tilde{\mathcal{F}}$ is first relaxed with a sequence of nonconvex functionals defined in $H^1\left(\mathrm{\Omega }\right)$ which, in turn, are discretized by finite elements. The $\mathrm{\Gamma }$-convergence of the discrete functionals to $\tilde{\mathcal{F}}$ as well as the compactness of any sequence of discrete absolute minimizers are proven. ...