arXiv subjects
Publications and source records attributed to Yong Luo.
In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$Δu+cu^α=0,$$ where $c, α$ are two real constants and $c\neq 0$.
In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in \cite{Luo} to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in $\mathbb{S}^5$, and then we use this relation to prove a classification result for Willmore Legendrian spheres in $\mathbb{S}^5$. We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in $\mathbb{S}^5$ belongs to $[0,2]$, then it must either be $0$ and $L$ is totally geodesic or $2$ and $L$ is a flat minimal Legendrian tori, which generalizes a result of \cite{YKM}. We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let $Σ$ be a closed surface and $(M,α,g_α,J)$ a 5-dimensional Sasakian manifold with a contact form $α$, an associated metric $g_α$ and an almost complex structure $J$. Assume that $f:Σ\mapsto M$ is a Legendrian immersion. Then $f$ is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if $(M,α,g_α,J)$ is a Sasakian Einstein manifold, in particular $\mathbb{S}^5$.
Let $(M^5,α,g_α,J)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $α$, associated metric $g_α$ and almost complex structure $J$ and $L$ a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the following equation \begin{eqnarray}\label{equ} -Δ^νH+(K-1)H=0, \end{eqnarray} where $Δ^ν$ is the normal Laplacian w.r.t the metric $g$ on $L$ induced from $g_α$ and $K$ is the Gauss curvature of $(L,g)$. Using equation \eqref{equ} and a new Simons' type inequality for Legendrian surfaces in the standard unit sphere $\mathbb{S}^5$, we prove an integral inequality for contact stationary Legendrian surfaces in $\mathbb{S}^5$. In particular, we prove that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$, $B$ is the second fundamental form of $L$, $S=|B|^2$, $ρ^2=S-2H^2$ and $$0\leq S\leq 2,$$ then we have either $ρ^2=0$ and $L$ is totally umbilic or $ρ^2\neq 0$, $S=2, H=0$ and $L$ is a flat minimal Legendrian torus.
Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary and $(N,h)$ be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from $M$ to $N$ under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from $M$ to $N$ under any Dirichlet boundary condition.
In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $ϕ:(M,g)\to (N, h)$ is a biharmonic map, where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian manifold with nonpositive sectional curvature, we will prove that $ϕ$ is a harmonic map if one of the following conditions holds: (i) $|dϕ|$ is bounded in $L^q(M)$ and $ \int_M|τ(ϕ)|^pdv_g<\infty, $ for some $1\leq q\leq\infty$, $1< p<\infty$; or (ii) $Vol(M)=\infty$ and $ \int_M|τ(ϕ)|^pdv_g<\infty, $ for some $1< p<\infty$. In addition if $N$ has negative sectional curvature, we assume that $rankϕ(q)\geq2$ for some $q\in M$ and $\int_M|τ(ϕ)|^pdv_g<\infty, $ for some $1< p<\infty$. These results improve the related theorems due to Baird et al.(cf. \cite{BFO}), Nakauchi et al.(cf. \cite{NUG}), Maeta(cf. \cite{Ma}) and Luo(cf. \cite{Luo}).
In this note we consider the Liouville type theorem for a properly immersed submanifold $M$ in a complete Riemmanian manifold $N$. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\geq-L(1+dist_N(\cdot,q_0)^2)^\fracα{2}$ for some $L>0, 2>α\geq 0$ and $q_0\in N$. (i) If $Δ|\vec{H}|^{2p-2}\geq k|\vec{H}|^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal. (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $Δu\geq ku^a$ for some constant $k>0$ and $a>1$. If $|\vec{H}|\leq C(1+dist_N(\cdot,q_0)^2)^\fracβ{2}$ for some $C>0$, $0\leqβ<1$, then $u=0$ on $M$. As applications we get some nonexistence result for $p$-biharmonic submanifolds.
Canonical correlation analysis (CCA) has proven an effective tool for two-view dimension reduction due to its profound theoretical foundation and success in practical applications. In respect of multi-view learning, however, it is limited by its capability of only handling data represented by two-view features, while in many real-world applications, the number of views is frequently many more. Although the ad hoc way of simultaneously exploring all possible pairs of features can numerically deal with multi-view data, it ignores the high order statistics (correlation information) which can only be discovered by simultaneously exploring all features. Therefore, in this work, we develop tensor CCA (TCCA) which straightforwardly yet naturally generalizes CCA to handle the data of an arbitrary number of views by analyzing the covariance tensor of the different views. TCCA aims to directly maximize the canonical correlation of multiple (more than two) views. Crucially, we prove that the multi-view canonical correlation maximization problem is equivalent to finding the best rank-1 approximation of the data covariance tensor, which can be solved efficiently using the well-known alternating least squares (ALS) algorithm. As a consequence, the high order correlation information contained in the different views is explored and thus a more reliable common subspace shared by all features can be obtained. In addition, a non-linear extension of TCCA is presented. Experiments on various challenge tasks, including large scale biometric structure prediction, internet advertisement classification and web image annotation, demonstrate the effectiveness of the proposed method.
In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y. L. Ou and L. Tang in \cite{Ou-Ta}. However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: 1. Any complete biharmonic submanifold (resp. hypersurface) $(M, g)$ in a Riemannian manifold $(N, h)$ with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some $p\in (0, +\infty)$, $\int_{M}|\vec{H}|^{p}du_g<+\infty,$ where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow N$, must be minimal. This generalizes the recent results due to N. Nakauchi and H. Urakawa in \cite{Na-Ur1} and \cite{Na-Ur2}. 2. Any complete biharmonic submanifold (resp. hypersurface) in a Reimannian manifold of at most polynomial volume growth whose sectional curvature (resp. Ricci curvature) is non-positive must be minimal. 3. Any complete biharmonic submanifold (resp. hypersurface) in a non-positively curved manifold whose sectional curvature (resp. Ricci curvature) is smaller that $-ε$ for some $ε>0$ which satisfies that $\int_{B_ρ(x_0)}|\vec{H}|^{p+2}dμ_g(p\geq0)$ is of at most polynomial growth of $ρ$, must be minimal. We also consider $\varepsilon$-superbiharmonic submanifolds defined recently in \cite{Wh} by G. Wheeler and prove similar results for $\varepsilon$-superbiharmonic submanifolds, which generalize the result in \cite{Wh}.
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. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositive curved space forms are minimal.
We prove the removal singularity results for maps with bounded energy from the unit disk $B$ of $R^2$ centered at the origin to a closed Riemannian manifold whose tension field is unbounded in $L^2(B)$ but satisfies the following condition: {eqnarray*} (\int_{B_t\setminus B_{\frac{t}{2}}}|τ(u)|^2)^1/2\leq C_1(\frac{1}{t})^a, {eqnarray*} for some $0<a<1$ and for any $0<t<1$, where $C_1$ is a constant independent of $t$. We will also prove that if a sequence $\{u_n\}$ has uniformly bounded energy and satisfies {eqnarray*} (\int_{B_t\setminus B_{\frac{t}{2}}}|τ(u_n)|^2)^1/2\leq C_2(\frac{1}{t})^a, {eqnarray*} for some $0<a<1$ and for any $0<t<1$, where $C_2$ is a constant independent of $n$ and $t$, then the energy identity holds for this sequence and there will be no neck formation during the blow up process.
In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.
In this note we will prove that an $n$ dimensional graphic self-shrinker in $R^{n+m}$ with flat normal bundle is a linear subspace. This result is a generalization of the corresponding result of Lu Wang in codimension one case.
Let $a>b>0$ and $f$ be a conformal map from $B_a\setminus B_b\subseteq R^2$ into $\R^n$, with $|\nabla f|^2=2e^{2u}$. Then $(e_1, e_2)$ with $e_1=e^{-u}\frac{\partial f}{\partial r},$ and $e_2=r^{-1}e^{-u}\frac{\partial f}{\partialθ}$ is a moving frame on $f(B_a\setminus B_b)$. It satisfies the following equation $$d\star =0,$$ where $\star$ is the Hodge star operator on $R^2$ with respect to the standard metric. We will study the Dirichret energy of this frame and give some applications.
In this paper, we prove the strong normalisation for Martin-Löf's Logical Framework, and suggest that {}``correct arity'', a condition weaker than well-typedness, will also guarantee the strong normalisation.