Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$ . Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb {S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \le 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$ , we obtain a smooth minimizer $f:{\mathbb {S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$ , where $H$ is the mean curvature.     

Approximations of periodic functions to \mathbb R ^n by curvatures of closed curves

We show that for any $n$ real periodic functions $f_1,\ldots , f_n$ with the same period, such that $f_i>0$ for $i<n$ , and a real number $\varepsilon >0$ , there is a closed curve in $\mathbb R ^{n+1}$ with curvatures $\kappa _1, \ldots , \kappa _n$ such that $\left| \kappa _{i(t)}-f_{i(t)}\right|<\varepsilon $ for all $i$ and $t$ . This does not hold for parametric families of closed curves in $\mathbb R ^{n+1}$ .     

Existence of integral m-varifolds minimizing \int \!|A|^p and \int \!|H|^p,\,p>m, in Riemannian manifolds

We prove existence of integral rectifiable $m$ -dimensional varifolds minimizing functionals of the type $\int |H|^p$ and $\int |A|^p$ in a given Riemannian $n$ -dimensional manifold $(N,g),\,2\le m<n$ and $p>m,$ under suitable assumptions on $N$ (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in ${\mathbb{R }^S}$ involving $\int |H|^p,$ to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.     

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.     

Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in {\varvec{R}}^{2n}

In this paper, let $n$ be a positive integer and $P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )$ for some integer $\kappa \in [0, n]$ , we prove that for any compact convex hypersurface $\Sigma $ in $\mathbf{R}^{2n}$ with $n\ge 2$ there exist at least two geometrically distinct P-invariant closed characteristics on $\Sigma $ , provided that $\Sigma $ is P-symmetric, i.e., $x\in \Sigma $ implies $Px\in \Sigma $ . This work is shown to extend and unify several earlier works on this subject.     

Geometry of the Funk metric on Weil–Petersson spaces

We discuss the Funk function $F(x,y)$ on a Teichmüller space with its Weil–Petersson metric $(\mathcal{T },d)$ introduced in Yamada (Convex bodies in Euclidean and Weil–Petersson geometries, 2011), which was originally studied for an open convex subset in a Euclidean space by Funk [cf. Papadopoulos and Troyanov (Math Proc Cambridge Philos Soc 147:419–437, 2009)]. $F(x,y)$ is an asymmetric distance and invariant by the action of the mapping class group. Unlike the original one, $F(x,y)$ is not always convex in $y$ with $x$ fixed (Corollary 2.11, Theorem 5.1). For each pseudo-Anosov mapping class $g$ and a point $x \in \mathcal{T }$ , there exists $E$ such that for all $n\not = 0$ , $ \log |n| -E \le F(x,g^n.x) \le \log |n|+E$ (Corollary 2.10), while $F(x,g^n.x)$ is bounded if $g$ is a Dehn twist (Proposition 2.13). The translation length is defined by $|g|_F=\inf _{x \in \mathcal{T }}F(x,g.x)$ for a map $g: \mathcal{T }\rightarrow \mathcal{T }$ . If $g$ is a pseudo-Anosov mapping class, there exists $Q$ such that for all $n \not = 0$ , $\log |n| -Q \le |g^n|_F \le \log |n| + Q.$ For sufficiently large $n$ , $|g^n|_F >0$ and the infimum is achieved. If $g$ is a Dehn twist, then $|g^n|_F=0$ for each $n$ (Theorem 2.16). Some geodesics in $(\mathcal{T },d)$ are geodesics in terms of $F$ as well. We find a decomposition of $\mathcal{T }$ by sets, each of which is foliated by those geodesics (Theorem 4.10).     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Some properties of the Yamabe soliton and the related nonlinear elliptic equation

Firstly we prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot \nabla u=0$ in $\mathbb{R }^{n}$ when $n\ge 3$ , $0<m\le \frac{n-2}{n}$ , $\alpha <0$ and $\beta \le 0$ and prove various properties of the solution of the above elliptic equation for other parameter range of $\alpha $ and $\beta $ . Then these results are applied to prove some results on Yamabe solitons including the exact behaviour of the metric of the Yamabe soliton, its scalar curvature and sectional curvature, at infinity. A new proof of a result of Daskalopoulos and Sesum (The classification of locally conformally flat Yamabe solitons, http://arxiv.org/abs/1104.2242) on the positivity of the sectional curvature of Yamabe solitons is also presented.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

A characterization of {\square(\kappa^{+})} in extender models

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a ${\square(\kappa^{+})}$ -sequence if and only if there is a pair of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ which, in turn, is equivalent to the existence of a family of size ${ < \kappa}$ of stationary subsets of ${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$ without common reflection point of cofinality ${ < \kappa}$ . By a result of Burke/Jensen, ${\square_\kappa}$ fails whenever ${\kappa}$ is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical ${\square(\kappa^{+})}$ -sequence where ${\kappa}$ is the first subcompact.     

Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(n\ge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>\sigma _{n}K_{\max }$ , where $\sigma _n\in (\frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(\ge {\!\!4})$ -dimensional compact Einstein manifold satisfies $K_{\min }\ge \eta _n R_0,$ where $\eta _n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $\eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(\ge {\!\!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $c\ge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.     

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke–Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda $ . It is shown that $\left||f \right||_\infty \ll _{\lambda ,\epsilon } N^{-\frac{1}{6}+\epsilon } \left||f \right||_2$ for any $\epsilon >0$ .     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

Equivariant entire solutions to the elliptic system \Delta u=W_u(u) for general G-invariant potentials

We consider the system $\Delta u - W_u(u) = 0$ , where $u: \mathbb R ^n \rightarrow \mathbb R ^m$ , for potentials $W: \mathbb R ^m \rightarrow \mathbb R $ that possess $N$ global minima and are invariant under a finite reflection group $G$ . We prove the existence of nontrivial $G$ -equivariant entire solutions connecting the $N$ minima of $W$ . Our proof only requires the minima of $W$ to be nondegenerate and an assumption on the behavior of $W$ for large $u$ .     

Reducibility and Unitarily Equivalence for a Class of Analytic Multipliers on the Dirichlet Space

In this paper, we first prove that if $\phi $ is a finite Blaschke product with $N=2,3$ zeros, then $M_\phi $ is reducible on the Dirichlet space if and only if $\phi $ is equivalent to $z^N$ . Also, we prove that $M_\phi $ is unitary equivalent to Dirichlet shift of multiplicity $N$ if and only if $\phi =\lambda z^N$ for some unimodular constant $\lambda $ .     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

Heegner cycles and higher weight specializations of big Heegner points

Let ${\mathbf{{f}}}$ be a $p$ -ordinary Hida family of tame level $N$ , and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis relative to $N$ . By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level $\Gamma _0(N)\cap \Gamma _1(p^s)$ , Howard has constructed a canonical class $\mathfrak{Z }$ in the cohomology of a self-dual twist of the big Galois representation associated to ${\mathbf{{f}}}$ . If a $p$ -ordinary eigenform $f$ on $\Gamma _0(N)$ of weight $k>2$ is the specialization of ${\mathbf{{f}}}$ at $\nu $ , one thus obtains from $\mathfrak{Z }_{\nu }$ a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes $\mathfrak{Z }_{\nu }$ to the étale Abel-Jacobi images of Heegner cycles when $p$ splits in $K$ .     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$