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SOME RESULTS OF EXPONENTIALLY BIHARMONIC MAPS INTO A NON-POSITIVELY CURVED MANIFOLD
Han, Yingbo Korean Mathematical Society 2016 대한수학회보 Vol.53 No.6
In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).
SOME RESULTS OF p-BIHARMONIC MAPS INTO A NON-POSITIVELY CURVED MANIFOLD
HAN, YINGBO,ZHANG, WEI Korean Mathematical Society 2015 대한수학회지 Vol.52 No.5
In this paper, we investigate p-biharmonic maps u : (M, g) $\rightarrow$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if ${\int}_M|{\tau}(u)|^{{\alpha}+p}dv_g$ < ${\infty}$ and ${\int}_M|d(u)|^2dv_g$ < ${\infty}$, then u is harmonic, where ${\alpha}{\geq}0$ is a nonnegative constant and $p{\geq}2$. We also obtain that any weakly convex p-biharmonic hypersurfaces in space formN(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen's conjecture for p-biharmonic submanifolds).
SOME RESULTS OF p-BIHARMONIC MAPS INTO A NON-POSITIVELY CURVED MANIFOLD
Yingbo Han,Wei Zhang 대한수학회 2015 대한수학회지 Vol.52 No.5
In this paper, we investigate p-biharmonic maps u : (M, g) → (N, h) from a Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. We obtain that if ∫M |τ(u)|a+pdvg < ∞ and ∫M |d(u)|2dvg < ∞, then u is harmonic, where a ≥ 0 is a nonnegative constant and p ≥ 2. We also obtain that any weakly convex p-biharmonic hypersurfaces in space form N(c) with c ≤ 0 is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen’s conjecture for p-biharmonic submanifolds).
Subgradient estimates for a nonlinear subelliptic equation on complete pseudohermitian manifold
Yingbo Han,Kaige Jiang,Mingheng Liang 대한수학회 2018 대한수학회보 Vol.55 No.1
Let $(M,J,\theta)$ be a complete pseudohermintian $(2n+1)$-mani\-fold. In this paper, we derive the subgradient estimate for positive solutions to a nonlinear subelliptic equation $\triangle_b u+a u\log u+b u=0$ on $M$, where $a\leq 0$, $b$ are two real constants.
Some results of exponentially biharmonic maps into a non-positively curved manifold
Yingbo Han 대한수학회 2016 대한수학회보 Vol.53 No.6
In this paper, we investigate exponentially biharmonic maps $u:(M,g)\rightarrow(N,h)$ from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if \begin{align*} &\int_Me^{\frac{p|\tau(u)|^2}{2}}|\tau(u)|^{p}dv_g<\infty~(p\geq2),~\int_M|\tau(u)|^{2}dv_g<\infty\text{ and}\\ &\int_M|du|^{2}dv_g<\infty, \end{align*} then $u$ is harmonic. When $u$ is an isometric immersion, we get that if $\int_Me^{\frac{pm^2|H|^2}{2}}|H|^{q}dv_g<\infty$ for $2\leq p<\infty$ and $0<q\leq p<\infty$, then $u$ is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form $N(c)$ with $c\leq 0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).
SUBGRADIENT ESTIMATES FOR A NONLINEAR SUBELLIPTIC EQUATION ON COMPLETE PSEUDOHERMITIAN MANIFOLD
Han, Yingbo,Jiang, Kaige,Liang, Mingheng Korean Mathematical Society 2018 대한수학회보 Vol.55 No.1
Let (M, J, ${\theta}$) be a complete pseudohermintian (2n+1)-manifold. In this paper, we derive the subgradient estimate for positive solutions to a nonlinear subelliptic equation ${\Delta}_bu+au{\log}u+bu=0$ on M, where $a{\leq}0$, b are two real constants.
LIOUVILLE THEOREMS FOR GENERALIZED SYMPHONIC MAPS
Feng, Shuxiang,Han, Yingbo Korean Mathematical Society 2019 대한수학회지 Vol.56 No.3
In this paper, we introduce the notion of the generalized symphonic map with respect to the functional ${\Phi}_{\varepsilon}$. Then we use the stress-energy tensor to obtain some monotonicity formulas and some Liouville results for these maps. We also obtain some Liouville type results by assuming some conditions on the asymptotic behavior of the maps at infinity.
Liouville theorems for generalized symphonic maps
Shuxiang Feng,Yingbo Han 대한수학회 2019 대한수학회지 Vol.56 No.3
In this paper, we introduce the notion of the generalized symphonic map with respect to the functional $\Phi_\varepsilon$. Then we use the stress-energy tensor to obtain some monotonicity formulas and some Liouville results for these maps. We also obtain some Liouville type results by assuming some conditions on the asymptotic behavior of the maps at infinity.