Emmanuel Humbert, Michel Vaugon
Let $(M,g)$ be a compact Riemannian manifold on dimension $n \geq 4$ not conformally diffeomorphic to the sphere $S^n$. We prove that a smooth function $f$ on $M$ is a critical function for a metric $\tilde{g}$ conformal to $g$ if and only if there exists $x \in M$ such that $f(x)>0$.
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We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.
In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constraint. All results are valid in the first Heisenberg group $\mathbb{H}^1$.
In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).
In this paper, we study the prescribed $Q$-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold $(M,g)$, which was introduced by S. Brendle in \cite{B2003}, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and $\int_M Qdμ< (n-1)!\Vol\left( S^n \right) $. In this paper we study the critical case that $\int_M Qdμ= (n-1)!\Vol\left( S^n \right)$, we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence resu…
Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions
本文证明了由环境函数规定平均曲率的毛细超曲面的紧致性定理，推广了Schätzle和Bellettini的结果，并扩展了曲率varifold定义至有向积分varifold，同时讨论了边界平均曲率规定情形。
We prove existence of compact spacelike hypersurfaces with prescribed k - curvature in de Sitter space, where the prescription function depends on both space and the tilt function.
Variational properties of the total inverse mean curvature in the plane under boundary constraints
该研究针对半平面内带指定边界的曲线，分析其总逆平均曲率的变分行为，明确了给定面积下临界点的存在性与强稳定性，并据此推导出局部极小性性质，丰富了平面曲线变分问题的相关结论。
In this paper we study rotational surfaces in the space $\mathbb{H}^2\times\mathbb{R}$ whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.
Consider a compact Lie group $G$ and a closed Lie subgroup $H<G$. Let $\mathcal M$ be the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. By studying variational properties of the scalar curvature functional on $\mathcal M$, we obtain an existence theorem for solutions to the prescribed Ricci curvature problem on $M$. To illustrate the applicability of this result, we explore cases where $M$ is a generalised Wallach space and a generalised flag manifold.
We obtain a classification result for rotational surfaces in the Heisenberg space and the universal cover of the special linear group, whose mean curvature is given as a prescribed $C^1$ function depending on their angle function. We show that these surfaces behave like the Delaunay surfaces of constant mean curvature, under some assumptions on the prescribed function. In contrast with the constant mean curvature case, we exhibit the existence of rotational, embedded tori, providing counterexamples of the Alexandrov problem for this class of immersed surfaces.
We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball $\mathbb{B}^n$ and on the boundary $\mathbb{S}^{n-1}$ the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case $n\geq 5$, under suitable pinching conditions on the mean curvature function.
It is well known that functions in involution with respect to Poisson brackets have a privileged role in the theory of completely integrable systems. Finding functionally independent functions in involution with a given function $h$ on a Poisson manifold is a fundamental problem of this theory and is very useful for the explicit integration of the equations of motion defined by $h$. In this paper, we present our results on the study of the inverse, so to speak, problem. By developing a technique analogous to that presented in P. Damianou and F. Petalidou, Poisson brackets with prescribed Casim…
We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including smooth Mor…
We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.
Remarks on prescribed scalar curvature functions
本文针对一般闭流形上带规定系数的Yamabe型半线性方程正解问题，给出最优存在性充分条件，并将其应用于标量平坦流形，解决了规定标量曲率问题中的一个长期疑问。
We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the classification of rotational surfaces of constant Gauss curvature; exhibit examples that cannot exist in the constant Gauss curvature case; and analyze the asymptotic behavior of strictly convex graphs. We also prove the existence of singular radial solutions intersecting orthogonally the axis of rotation.
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
Prescribed $L_{p}$ curvature problem for convex capillary hypersurface
本文研究了欧几里得半空间中凸毛细超曲面的 prescribed $L_p$ 曲率问题，通过转化问题为 Hessian 商类型方程，证明了解的存在性和唯一性，并应用该方法解决了毛细 $L_p$ Christoffel-Minkowski 问题及相关特征值问题。
Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature
针对单位三维球面上满足挤压条件的正函数或变号函数h，证明存在至少两个具有预设平均曲率h的嵌入二维球面，拓展了预设平均曲率嵌入球面的存在性结果。
Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed incompressible surface with mean curvature $H$ . A direct application is the existence of embedded surfaces of prescribed constant mean curvatures with constants in $(-2,2)$.
In this paper we solve the Björling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in $\mathbb{R}^3$
We construct families of smooth functions $H\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that the Euclidean $(n+1)$-space is completely filled by not necessarily round hyperspheres of mean curvature $H$ at every point.
Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds
该研究建立了非紧流形中预定平均曲率超曲面的极小-极大理论，适用于紧集外不变号的预定函数，并借此证明了两类超曲面的新存在性结果。
On the variational properties of the prescribed Ricci curvature functional
本文研究齐次度量的预定 Ricci 曲率问题，通过考察标量曲率泛函的全局性质，为一般齐次空间给出了该泛函存在全局最大值的条件，并通过具体例子加以说明。
Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature
研究在紧致黎曼流形上寻找共形度量，使得内部标量曲率为零且边界平均曲率为指定函数，通过构造局部测试函数解决了Escobar工作中大部分未解决问题并建立了新的可解性条件。
The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(κ)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.
Prescribing scalar curvatures: non compactness versus critical points at infinity
研究通过例子展示在黎曼流形上，标量曲率的规范问题中，非紧性与无穷远处临界点的转变现象。
Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree
本文研究具有负Gauduchon度的紧Hermitian流形上预定Chern数量曲率问题，利用几何流收敛性在非正及变号曲率函数情形下建立了存在性与不存在性结果。
We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function $f$ as its scalar curvature.
In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.
Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds
本文证明了在无边界光滑黎曼流形中，函数非退化临界点附近存在由平均曲率与该函数成比例的球面构成的唯一叶状结构，并指出放宽非退化假设会丧失该叶状结构性质。
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.
Fully nonlinear elliptic equations for some prescribed curvature problems on Hermitian manifolds
该研究利用爆破论证与部分一致椭圆性研究Hermitian流形上的完全非线性椭圆方程，将其应用于寻找指定Chern-Ricci曲率的共形Hermitian度量的几何问题，且假设因几何函数理论阻碍几乎达最优。
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive consta…
In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives estimates on the Morse index of the solutions and has the advantage to extend known existence results. Moreover it provides, for generic $K$ {\it Morse Inequalities at Infinity}, which give a lower bound on the numb…
We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type classification holds for this class of hypersurfaces. We also exhibit examples showing that the behavior of rotational hypersurfaces of prescribed (non-constant) mean curvature is much richer than in the constant mean curvature case.
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 pro…
We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.
Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components of $\partial A$. This variational problem is a problem with lack of compactness hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf differential and on a strong link between this problem and the problem of finding all minima…
We consider a free boundary problem for the Willmore functional. Given a smooth domain $Ω$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial Ω$ orthogonally along their boundary and having small prescribed area. Using rescaling we first obtain constrained solutions with prescribed two-dimensional barycenter, and then study the variation of the barycenter.
Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $ω$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.
Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\int_M f dμ_{{\it g}_0} <0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation \[ \mathbf{P}_{g_0} u = f e^{4u}, \] where $\mathbf{P}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, we…
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
Obstructions to prescribed Q-curvature of complete conformal metrics on $\mathbb{R}^n$
本文研究了 $\mathbb{R}^n$ 上完备共形度量的预定 Q-曲率问题，给出了一个 Bonnet-Mayer 型障碍定理，并揭示了预定函数衰减速率带来的其他障碍。
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