Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms ${M or_P (\mathbb{G}, M(n, \xi))}$ and the moduli space of stable bundles over ${X \times \mathbb{G}}$ , where ${\mathbb{G}}$ is the Grassmannian ${\mathbb{G}(n - r, \mathbb{C}^n)}$ . Moreover, we give sufficient conditions for ${M or_{2ns}(\mathbb{P}^1, M(n, \xi))}$ to be non-empty, when s ≥ 1.     

On the Schröder equation and iterative sequences of C r diffeomorphisms in {\mathbb{R}^{N}} space

Let ${U \subset \mathbb{R}^{N}}$ be a neighbourhood of the origin and a function ${F:U\rightarrow U}$ be of class C ^r , r ≥ 2, F(0) = 0. Denote by F ⁿ the n-th iterate of F and let ${0<|s_1|\leq \cdots \leq|s_N| <1 }$ , where ${s_1, \ldots , s_N}$ are the eigenvalues of dF(0). Assume that the Schröder equation ${\varphi(F(x))=S\varphi(x)}$ , where S: = dF(0) has a C ² solution φ such that dφ(0) = id. If ${\frac{log|s_1|}{log|s_N|} <2 }$ then the sequence {S ^?n F ⁿ (x)} converges for every point x from the basin of attraction of F to a C ² solution φ of (1). If ${2\leq\frac{log|s_1|}{log|s_N|} }$ then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S ^?n F ⁿ (x)}. Moreover, we show that if F is of class C ^r and ${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$ then every C ^r solution of the Schröder equation such that dφ(0) = id is given by the formula $$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$ where ${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$ are some homogeneous polynomials of degree k, which are determined by the differentials d ^(j) F(0) for 1 < j ≤  p.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

Hofer geometry of a subset of a symplectic manifold

To every closed subset X of a symplectic manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham (X, ω). We equip this group with a semi-norm ${\Vert\cdot\Vert^{X, \omega}}$ , generalizing the Hofer norm. We discuss Ham (X, ω) and ${\Vert\cdot\Vert^{X, \omega}}$ if X is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in ${\mathbb{R}^{2n}}$ this diameter is bounded below by ${\frac{\pi}{2}}$ , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in ${\mathbb{R}^{2n}}$ , such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π / k(n, d), where k(n, d) is an explicitly defined integer.     

Some Critical Almost Hermitian Structures

Let M be an even dimensional compact smooth manifold admitting an almost complex structure. Let ${{(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)}$ . We discuss the critical points of the functional ${\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g}$ on the space of all almost Hermitian structures ${\mathcal{AH}(M)}$ on M and its subspace ${{\mathcal{AH}_{c}(M)}}$ with a certain positive constant c, where τ and τ ^* are the scalar curvature and the *-scalar curvature of (J, g), respectively. Further, we provide some examples illustrating our arguments.     

Critical behavior and the limit distribution for long-range oriented percolation. I

We consider oriented percolation on ${\mathbb{Z}}^d\times{\mathbb{Z}}_+$ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on ${\mathbb{Z}}^d$ . Suppose that D(x) decays as |x|^?d?α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension $d_c=2(\alpha\wedge2)$ . We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to $e^{-c|k|^{\alpha\wedge2}}$ for some c > 0.     

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

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).     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Integration of H?lder forms and currents in snowflake spaces

For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral ${\int_M f \, dg_1 \wedge \cdots \wedge dg_n}$ in case the functions ${f, g_1, \ldots, g_n}$ are merely H?lder continuous of a certain order by extending the construction of the Riemann?CStieltjes integral to higher dimensions. More generally, we show that for ${\alpha \in (\tfrac{n}{n+1},1]}$ the n-dimensional locally normal currents in a locally compact metric space (X, d) represent a subspace of the n-dimensional currents in (X, d ^?? ). On the other hand, for ${n \geq 1}$ and ${\alpha \leq \tfrac{n}{n+1}}$ the vector space of n-dimensional currents in (X, d ^?? ) is zero.     

On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space

We study the Laplace equation in the half-space ${\mathbb{R}_{+}^{n}}$ with a nonlinear supercritical Robin boundary condition ${\frac{\partial u}{\partial\eta }+\lambda u=u\left\vert u\right\vert^{\rho-1}+f(x)}$ on ${\partial \mathbb{R}_{+}^{n}=\mathbb{R}^{n-1}}$ , where n ≥ 3 and λ ≥ 0. Existence of solutions ${u \in E_{pq}= \mathcal{D}^{1, p}(\mathbb{R}_{+}^{n}) \cap L^{q}(\mathbb{R}_{+}^{n})}$ is obtained by means of a fixed point argument for a small data $f \in {L^{d}(\mathbb{R}^{n-1})}$ . The indexes p, q are chosen for the norm ${\Vert\cdot\Vert_{E_{pq}}}$ to be invariant by scaling of the boundary problem. The solution u is positive whether f(x) > 0 a.e. ${x\in\mathbb{R}^{n-1}}$ . When f is radially symmetric, u is invariant under rotations around the axis {x _n  = 0}. Moreover, in a certain L ^q -norm, we show that solutions depend continuously on the parameter λ ≥ 0.     

Computable representations for convex hulls of low-dimensional quadratic forms

Let ${\mathcal{C}}$ be the convex hull of points ${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$ . Representing or approximating ${\mathcal{C}}$ is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and ${\mathcal{F}}$ is a simplex, then ${\mathcal{C}}$ has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and ${\mathcal{F}}$ is a box, then ${\mathcal{C}}$ has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ when ${\mathcal{F}\subset\Re^2}$ is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ . When n = 3 and ${\mathcal{F}}$ is a box, we show that a representation for ${\mathcal{C}}$ can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Monotonicity and logarithmic convexity relating to the volume of the unit ball

Let Ω _n denote the volume of the unit ball in ${\mathbb{R}^n}$ for ${n\in\mathbb{N}}$ . In the present paper, the authors prove that the sequence ${\Omega_{n}^{1/(n\,{\rm ln}\,n)}}$ is logarithmically convex and that the sequence ${\frac{\Omega_{n}^{1/(n\,{\rm ln}\,n)}}{\Omega_{n+1}^{1/[(n+1)\,{\rm ln}(n+1)]}}}$ is strictly decreasing for n ≥ 2. In addition, some monotonic and concave properties of several functions relating to Ω _n are extended and generalized.     

Wintgen Ideal Surfaces in Four-dimensional Neutral Indefinite Space Form {R^4_2(c)}

For an oriented space-like surface M in a four-dimensional indefinite space form ${R^4_2(c)}$ , there is a Wintgen type inequality; namely, the Gauss curvature K, the normal curvature K ^D and mean curvature vector H of M in ${R^4_2(c)}$ satisfy the general inequality: ${K+K^D \geq \langle H,H \rangle+c}$ . An oriented space-like surface in ${R^4_2(c)}$ is called Wintgen ideal if it satisfies the equality case of the inequality identically. In this paper, we study Wintgen ideal surfaces in ${R^4_2(c)}$ . In particular, we classify Wintgen ideal surfaces in ${R^4_2(c)}$ with constant Gauss and normal curvatures. We also completely classify Wintgen ideal surfaces in ${\mathbb E^4_2}$ satisfying |K| = |K ^D | identically.