We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparametrizations. In particular we investigate the metric, for a constant $A>0$, $G_c^A(h,k):={\int }_{S^1}(1+A{\kappa }_c{\left(\theta \right)}^2)\langle h\left(\theta \right),k\left(\theta \right)\rangle \left|c^{\text{'}}\left(\theta \right)\right|d\theta$ where ${\kappa }_c$ is the curvature of the curve $c$ and $h$, $k$ are normal vector fields to $c$. The term $A{\kappa }^2$ is a sort of geometric Tikhonov regularization because, for $A=0$, the geodesic distance between any two distinct curves is 0, while...

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...

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.

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...

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 $$\left\{\begin{array}{c}{f}^{\text{'}\text{'}}\left(t\right)+k\left(t\right)f\left(t\right)=0\text{on}(0,\infty ),\hfill \\ f\left(0\right)=0,f^{\text{'}}\left(0\right)=1.\hfill \end{array}\right.$$

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 $n=1$....

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.

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is...

We establish in this paper a lower bound for the volume of a unit vector field $\overrightarrow{v}$ defined on ${𝐒}^n\setminus \{\pm x\}$, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\overrightarrow{v}$ at $x$ and $-x$.

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.

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...

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 ${\S }^{N-2}$.

We determine in ${ℝ}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $\Omega$ of affinities containing the translation group is a geodesic with respect to $\nabla$. Special attention is paid to the case that $\Omega$ contains many dilatations or that $C$ is a curve in ${ℝ}^3$. If $C$ is a curve in ${ℝ}^3$ and $\Omega$ is the translation group then we calculate not only the components of the curvature and the Weyl...