Sun-Yung Alice Chang, Hao Fang
We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of the metric on the manifold.
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Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the immersion $S^2\to\mathbb{R}^3$.
In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change then surface is minimal. Also, it is determined that the critical point of mean curvature functional is homeomorphic to the sphere. Furthermore, the Euler-Lagrange equations associated to the mean curvature and Willmore functionals are determined.
The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of governing differential equations for variational problems on the space of conformal vector fields. Several examples are given. In addition, the paper also gives a review, in full detail, of already known vector field decompositions involving subalgebras of volume preserving and symp…
Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particular for $K$ a negative constant, this provides an elementary proof of the uniformization theorem for compact Riemann surfaces of genus $g >1$.
We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $θ$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJθ$, where $J$ denotes the complex structure. These extremal metr…
We define a new formal Riemannian metric on a conformal class in the context of the $v_{\frac{n}{2}}$-Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in a given conformal class.
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal…
In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum within its conformal class, unless the manifold is isometric to round sphere with the standard metric up to a multiple of constant. Note that when $(M,g)$ is locally conformally flat, this functional reduces to the well-studied $\int_M σ_3(g)dv_g$. Hence, our result generalize a pr…
We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_1.
In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and relies on the reduction theory for coisotropic optimal control problems. This gives a unified explanation of the integrability of several classical variational problems such as the total squared curvature functional, the projective, conformal and pseudo-conformal arc-length funct…
In this paper, we introduce a new functional for the conformal spectrum of the conformal laplacian on a closed manifold M of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds.
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting…
A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants
本文针对共形变分标量黎曼不变量，给出流形承认多非同伦及无穷多周期共形重缩放使不变量为常数的充分条件，改进并建立了各阶Q-曲率及重整化体积系数的非唯一性结果。
Conformal geodesics are solutions to a system of third order of equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third--order conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth order ODEs arising from this Lagrangian, and show that some of its integral curves are spirals.
Conformal geodesics are not variational in higher dimensions
本文证明参数化和未参数化共形测地线方程在高于三维时均不能由共形不变泛函的 Euler-Lagrange 方程导出，提示变分原理或可选择物理维度。
We consider the problem of minimizing the Willmore energy in the class of conformal immersions of a given closed, genus p Riemann surface into R^n for n=3,4. We prove existence of a smooth minimizer, provided that the infimum is below a certain bound ${\cal W}(n,p)$. For tori in R^3 we have explicitely ${\cal W}(3,1) = 8π$.
Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $Φ(f)$ $=$ ${\displaystyle \int_M |T_f|^2dv_g}$. We call such maps {\it C-stationary maps}. …
In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $n\ge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.
Conformal hypersurface invariants and Bach-type Boundary Problems
本文利用变分方法发现偶数维共形流形中一个新的对称无迹张量超曲面共形不变量，证明其完善了共形基本形式族，并据此解决了Poincaré--Einstein问题中的Dirichlet--to--Neumann映射像问题，同时构造了边界的新全局不变量及Bach平坦流形的度量。
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functional on the manifold.
Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short introduction to super differential geometry we will compare Riemann surfaces and super Riemann surfaces. We will see that super Riemann surfaces can be viewed as Riemann surfaces with an additional field, the gravitino. An extension of the harmonic action functional to super Riemann s…
We analyze an energy functional associated to Conformal Ricci Flow along closed manifolds with constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of both the metric and the pressure function along Conformal Ricci Flow.
We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise $(k,r)$-jet. The relations to the Euler-Lagrange morphism of the variational calculus are described.
We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e β^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)β^2(e),$$ where $β(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $β$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are obse…
On the conformal-biharmonic stability of the identity map of Einstein manifolds
研究爱因斯坦流形恒等映射在共形双能功能下的稳定性，发现其与谐波稳定性指数相同，唯一体育四维欧几里得球面例外，该映射在能量功能下不稳定，在共形双能功能下稳定。
We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in ${\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact mani…
We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on $M$.
Quaternionic Analysis of Conformal Maps and the Willmore Functional
本文将四元数分析推广到弱共形映射，证明Willmore泛函在紧黎曼曲面到三维或四维欧氏空间的弱共形映射空间中达到最小值，并研究其正则性。
A Variational Approach to the Yamabe Problem: Conformal Transformations and Scalar Curvature on Compact Riemannian Manifolds
本文用变分与集中紧性方法研究紧Riemann流形上的Yamabe问题，证明在$\lambda(M)<\lambda(S^n)$条件下存在共形常标量曲率度量。
Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.
A Restricted Chen-Nagano Variational Principle for the Einstein-Hilbert Functional
本文引入紧致黎曼流形上爱因斯坦-希尔伯特泛函的限制性陈-长野变分原理，通过陈-长野规范约束限制变分问题，推导出限制性欧拉-拉格朗日方程，得到临界度量的新结构刻画，并建立全局刚性结果。
The Fefferman Metric for Twistor CR Manifolds and Conformal Geodesics in Dimension Three
本文显式描述了扭量CR流形上的费弗曼度量，并证明链和零链投影到共形测地线，且任何共形测地线有提升，从而给出三维共形测地线的变分刻画。
The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.
Weighted isoperimetric ratios and extension problems for fractional conformal Laplacians
本研究建立了加权等周问题与分数阶共形拉普拉斯算子延拓问题的关联，推导了半空间与单位球上加权泊松积分的精确不等式并分类极值，构造CCE流形上的加权等周变分问题并证明其与Chang-González延拓的联系，给出极值存在准则并提出猜想。
In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in $R^m$, away possibly from isolated branched points, which minimizes the Willmore …
In this paper we investigate the nature of stationary points of functionals on the space of Riemannian metrics on a smooth compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric, and the manifold is the standard n-sphere, we apply methods from representation theory to give a universal form of the Hessian of the functional at a stationary point. This reveals a very strong rigidity in the local structure of any such functio…
Generalized Monge-Ampère functionals and related variational problems
该研究引入一族实值 Monge-Ampère 泛函，通过建立 Sobolev 型不等式分析其变分性质，并据此证明相关 Dirichlet 问题及非线性特征值问题解的存在性与唯一性。
By a conformal string in Euclidean space is meant a closed critical curve with non-constant conformal curvatures of the conformal arclength functional. We prove that (1) the set of conformal classes of conformal strings is in 1-1 correspondence with the rational points of the complex domain $\{q\in \mathbb{C} \,:\, 1/2 < \mathrm{Re}\, q < 1/\sqrt{2},\,\, \mathrm{Im}\, q > 0,\,\, |q| < 1/\sqrt{2}\}$ and (2) any conformal class has a model conformal string, called symmetrical configuration, which is determined by three phenomenological invariants: the order of its symmetry group and its linking …
A Variational Scalar Conformal Flow for Lorentz-Contracted Geometry: Algebraic Decay and Canonical Normalization
本文提出一种变分标量共形流，通过代数衰减律实现能量泛函的归一化，揭示了松弛算子的连续谱特性。
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $σ_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential oper…
This is a survey on rigidity and geometrization results obtained with the help of the discrete Hilbert-Einstein functional, written for the proceedings of the "Discrete Curvature" colloquium in Luminy.
The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection $A$ on a line bundle $L$ and a section $ϕ$ of another bundle $W^+$ constructed from $L$ and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.
The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3, g_ε)$ -where $g_ε$ is a metric close and asymptotic to the euclidean one. With the same technique a non existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.
A new functional for simplicial surfaces is suggested. It is invariant with respect to Moebius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. as an application a bending energy for discrete thin-shells is derived.
In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting opera…
A variational method for functionals depending on eigenvalues
该研究提出了一套针对依赖黎曼流形特征值泛函的系统变分方法，通过将极小极大方法推广到局部Lipschitz泛函构造Palais-Smale序列，并证明了二维下两类特征值组合对应序列的收敛性。
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.
We connect the well-known theory of functional forms of variational bicomplex with the theory of antiexact differential forms. We identify antiexact functional forms as an obstruction to the variationality of differential equations. The most prominent result of this observation is the formulation of the variational problem for some differential equations that are not variational and neither have a variational multiplier. Heat, Navier-Stokes, and KdV equations illustrate this new variational principle.
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.
We consider the reduced Allen-Cahn action functional, which appears as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of generalized hypersurfaces this functional consists of the sum of the squares of the mean curvature and of the velocity vector, integrated over time and space. For given initial and final conditions we investigate the corresponding action minimization problem. We give a generalized formulation and prove compactness and lower semicontin…
We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R^3. This hypothesis was formulated recently by I.A.Taimanov (dg-ga/9610013). Higher analogs of the Willmore functional are defined in terms of the Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with the zero-energy scattering problem for the two-dimensional Dirac operator.
A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrang…
While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvilinear quadrilaterals where three sides determine the fourth by a geometric "rectangle rule," which can be expressed also by functional formulas. There is an explicit functional degree of freedom in choosing the mapping taking the square to a given quadrilateral. We characterize cl…
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