Boundary partial regularity for a class of Dirac-harmonic maps
In this paper, we prove the boundary partial regularity for a class of coupled Dirac-harmonic maps satisfying a certain energy monotonicity inequality near the boundary.
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arXiv subjects
Jingyong Zhu
Publications and source records attributed to Jingyong Zhu.
Uniqueness of Dirac-harmonic maps from a compact surface with boundary
As a commutative version of the supersymmetric nonlinear sigma model, Dirac-harmonic maps from Riemann surfaces were introduced fifteen years ago. They are critical points of an unbounded conformally invariant functional involving two fields, a map from a Riemann surface into a Riemannian manifold and a section of a Dirac bundle which is the usual spinor bundle twisted with the pull-back of the tangent bundle of the target by the map. As solutions to a coupled nonlinear elliptic system, the existence and regularity theory of Dirac-harmonic maps has already received much attention, while the general uniqueness theory has not been established yet. For uncoupled Dirac-harmonic maps, the map components are harmonic maps. Since the uniqueness theory of harmonic maps from a compact surface with boundary is known, it is sufficient to consider the uniqueness of the spinor components, which are solutions to the corresponding boundary value problems for a nonlinear Dirac equation. In particular, when the map components belong to $W^{1,p}$ with $p>2$, the spinor components are uniquely determined by boundary values and map components. For coupled Dirac-harmonic maps, the map components are not harmonic maps. So the uniqueness problem is more difficult to solve. In this paper, we study the uniqueness problem on a compact surface with boundary. More precisely, we prove the energy convexity for weakly Dirac-harmonic maps from the unit disk with small energy. This yields the first uniqueness result about Dirac-harmonic maps from a surface conformal to the unit disk with small energy and arbitrary boundary values.
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Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds
Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds $M$ and $N$ were introduced in search of a natural notion of harmonicity for maps defined on a general even dimensional Riemannian manifold $M$. They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions. In this paper, we consider conformal-harmonic maps from certain locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.
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Dirac-harmonic maps with trivial index
For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with the map component in the given homotopy class. Most known results require the index to be nontrivial. When the index is trivial, the few known results are all constructive and produce uncoupled solutions. In this paper, we define a new quantity. As a byproduct of proving the homotopy invariance of this new quantity, we find a new simple proof for the fact that all Dirac-harmonic spheres in surfaces are uncoupled. More importantly, by using the homotopy invariance of this new quantity, we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case. In particular, when the domain is a closed Riemann surface, we prove the short-time existence of the $α$-Dirac-harmonic map flow in the trivial index case. Together with the density of the minimal kernel, we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to Kähler manifolds, which extends the previous result of the first and third authors. This establishes a general existence theory for Dirac-harmonic maps in the context of trivial index.
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Existence of Kazdan-Warner equation with sign-changing prescribed function
In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ \begin{align*} -Δu=8π\left(\frac{he^{u}}{\int_Σhe^{u}}-1\right) \end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8π}$ which is defined by \begin{align*} J_{8π}(u)=\frac{1}{16π}\int_Σ|\nabla u|^2+\int_Σu-\ln\left|\int_Σhe^{u}\right|,\quad u\in H^1\left(Σ\right). \end{align*} We prove the existence of minimizer of $J_{8π}$ by assuming \begin{equation*} Δ\ln h^++8π-2K>0 \end{equation*}at each maximum point of $2\ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{\varepsilon}$ of critical points of $J_{8π-\varepsilon}$ with $\int_Σhe^{u_{\varepsilon}}=1, \lim\limits_{\varepsilon\searrow 0}J_{8π-\varepsilon}\left(u_{\varepsilon}\right)<\infty$ and obtain the following identity during the blow-up process \begin{equation*} -\varepsilon=\frac{16π}{(8π-\varepsilon)h(p_\varepsilon)}\left[Δ\ln h(p_\varepsilon)+8π-2K(p_\varepsilon)\right]λ_{\varepsilon}e^{-λ_{\varepsilon}}+O\left(e^{-λ_{\varepsilon}}\right), \end{equation*}where $p_\varepsilon$ and $λ_\varepsilon$ are the maximum point and maximum value of $u_\varepsilon$, respectively. Moreover, $p_{\varepsilon}$ converges to the blow-up point which is a critical point of the function $2\ln h^{+}+A$.
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Uniqueness of Hypersurfaces of Constant Higher Order Mean Curvature in Hyperbolic Space
We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson to the embedded hypersurfaces with constant higher order mean curvature. Then we prove two Bernstein type results for immersed hypersurfaces under different assumptions. Last, we show the rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
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Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces
We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $α$-(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.
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Global existence and convergence of a flow to Kazdan-Warner equation with non-negative prescribed function
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Σ,g)$ \begin{align*} -Δ_{g}u=8π\left(\frac{he^{u}}{\int_Σhe^{u}{\rm d}μ_{g}}-\frac{1}{\int_Σ{\rm d}μ_{g}}\right) \end{align*} where the prescribed function $h\geq0$ and $\max_Σh>0$. We prove the global existence and convergence under additional assumptions such as \begin{align*} Δ_{g}\ln h(p_0)+8π-2K(p_0)>0 \end{align*} for any maximum point $p_0$ of the sum of $2\ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Σ$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145 (2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1 (1997), no. 2, 230-248] to the non-negative prescribed function case.
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Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space
The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in $\mathbb{H}^{n+1}$. Similar to the usual mean curvature flow, the MMCF is the natural negative $L^2$-gradient flow of the area-volume functional $\mathcal{I}(Σ)=A(Σ)+σV(Σ)$ associated to a hypersurface $Σ$. In this paper, we prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time. In general one cannot expect the convergence of the flow as it can be seen from the flow starting from a horosphere (whose asymptotic boundary is degenerate to a point).
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Short-time existence of the $α$-Dirac-harmonic map flow and applications
In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for $α$-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for $α$-Dirac-harmonic maps. The obstacle to the global existence is the singular time when the kernel of the Dirac operator no longer stays minimal along the flow. In this case, the kernel may not be continuous even if the map is smooth with respect to time. To overcome this issue, we use the analyticity of the target manifold to obtain the density of the maps along which the Dirac operator has minimal kernel in the homotopy class of the given initial map. Then, when we arrive at the singular time, this density allows us to pick another map which has lower energy to restart the flow. Thus, we get a flow which may not be continuous at a set of isolated points. Furthermore, with the help of small energy regularity and blow-up analysis, we finally get the existence of nontrivial $α$-Dirac-harmonic maps ($α\geq1$) from closed surfaces. Moreover, if the target manifold does not admit any nontrivial harmonic sphere, then the map part stays in the same homotopy class as the given initial map.
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$α$-Dirac-harmonic maps from closed surfaces
$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $α$-harmonic maps for $α>1$ and then letting $α\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $α$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $\varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $α$-Dirac-harmonic maps converges to a smooth nontrivial $α$-Dirac-harmonic map.
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Weakly Horospherically Convex Hypersurfaces in Hyperbolic Space
In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $ϕ:M^n \to \mathbb{H}^{n+1}$ and a class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some of the key aspects of the correspondence and its consequences have dimensional restrictions $n\geq3$ due to the reliance on an analytic proposition from [5] concerning the asymptotic behavior of conformal factors of conformal metrics on domains of $\mathbb{S}^n$. In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. In the case of a single point boundary $\partial_{\infty}ϕ(M)=\{x\} \subset \mathbb{S}^n$, we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in [2], we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from [2] to the case of surfaces in $\mathbb{H}^{3}$.
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A Sharp Height Estimate for the Spacelike Constant Mean Curvature Graph in the Lorentz-Minkowski Space
In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of critical point for the solution of such equation over convex domain, which is an analogue of the result in [28]. Last, by this uniqueness, we obtain a minimum principle for a functional depending on the solution and its gradient. This gives us a sharp gradient estimate for the solution, which leads to a sharp height estimate.
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Dirichlet Problem of Quaternionic Monge-Ampère Equations
In this paper, the author studies quaternionic Monge-Ampère equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to the open problem proposed by Semyon Alesker in [3], but also extends relevant results in [7] to the quaternionic vector space.
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