Julian Scheuer
These notes arose from a mini lecture series the author gave at the Early Career Researchers Workshop on Geometric Analysis and PDEs, held in January 2020 at the Matrix institute of the University of Melbourne. We discussed some classical aspects of expanding curvature flows and obtained first applications. In these notes we will give a detailed account on what was covered during the lectures.
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We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showing that extrinsic curvature, defined by a suitable pull-back of the volume form on the $n$-sphere via the Gauss map, is identical to intrinsic curvature, defined by the Pfaffian form. This latter fact is stated in form of an integral identity for the Brouwer degree of the Gauss m…
N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M$ contains no smooth, complete immersed surface with curvature below $K_1$. Optimal values of $K_1$ are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Amp{è}re equations.
We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we…
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …
Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.
We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.
Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic structure of the piecewise flat manifold and a choice of dual tessellation, and can be viewed as the average of $n$-volume integrals. The constructions are also independent of the manifold dimension.
We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.
In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira (Ann. Global Anal. Geom. 18, 371-383 (2000)). We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.
This paper defines two new extrinsic curvature quantities on the corner of a four-dimensional Riemannian manifold with corner. One of these is a pointwise conformal invariant, and the conformal transformation of the other is governed by a new linear second-order pointwise conformally invariant partial differential operator. The Gauss-Bonnet theorem is then stated in terms of these quantities.
Coarse extrinsic curvature of Riemannian submanifolds
本文受Ollivier粗Ricci曲率启发，提出基于Wasserstein 1-距离的黎曼子流形粗外曲率概念，并给出从Poisson点云等统计数据近似平均曲率的方法，可应用于流形学习与度量嵌入中的几何信息推断。
For any hypersurface of a Riemannian manifold, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we announce explicit formulas for the extrinsic Paneitz operators P_4 and the corresponding extrinsic Q-curvatures for totally umbilic hypersurfaces in any dimension. Moreover, we state explicit formulas for the critical extrinsic P_4 and the total integral of the critical extrinsic Q_4 in the general case. This integral is a global conformal invariant. Finally, we establish an analog of the Alexakis-Deser-Schwimmer decomposition of the critical …
Clifford structures, bilegendrian surfaces, and extrinsic curvature
本文用Clifford代数建立统一框架，将黎曼与半黎曼3-流形中的常外曲率浸入曲面对应于酉丛中的双勒让德曲面，并完整分类了3-球面单位切丛中的完备与紧致浸入双勒让德曲面。
From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships betwe…
Stable free boundary surfaces with constant extrinsic curvature in $3$-dimensional space forms
本文针对单连通三维空间形式测地球及R³平板中的自由边界H₂曲面，借助常高阶平均曲率自由边界曲面的稳定性概念，推导得出相应刚性结果。
Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds
本文证明了外蕴几何流在一般 ambient 流形中的逆唯一性定理，适用于多种几何流类型。
A Monotonicity Formula for the Extrinsic Biharmonic Map Heat Flow
该研究针对外在双调和映射热流，利用双调和热核性质推导出熵型量，并证明其在$n\leq 4$时具有单调性。
The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.
We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$surfaces). We establish that the only complete $K-$surfaces in $S^2\times R$ and $H^2\times R$ are rotational spheres. Here are the key steps to achieve this. First height estimates for compact $K-$surfaces in a general ambient space $M^2\times R$ with boundary in a slice are obta…
Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally-invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvat…
We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.
Analogous to the bowl soliton of mean curvature flow, we construct rotationally symmetric translating solutions to a very large class of extrinsic curvature flows, namely those whose speeds are $α$-homogeneous ($α>0$), elliptic and symmetric with respect to the principal curvatures. We show that these solutions are necessarily convex, and give precise criteria for the speed functions which determine whether these translators are defined on all of $\mathbb{R}^{n}$ or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we are also able t…
In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In the particular case of a surface of nonnegative curvature, we prove that the surface is diffeomorphic to the Euclidean plane if the surface has tamed second fundamental form, and that the surface is isometric to the Euclidean plane if the surface has strongly tamed second fundament…
Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications
针对完全非线性外蕴曲率流的平移孤子，发展旋转理论，获得碗型平移子的渐近展开，并引入符号颈框架构造和分类悬链面型平移子，进而证明严格凸整体图平移子的唯一性和有界图平移子的悬链面障碍不存在性。
Extrinsic Bonnet-Myers Theorem and almost rigidity
本文证明了正Ricci曲率紧致流形的外在Bonnet-Myers定理，并给出了欧氏空间中正截面曲率且外在直径几乎最大的紧致超曲面的几乎刚性结果。
The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a $1$-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in $\mathbb{R}^{n+1}$ for particular functions.
Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in $M$ centered at a point $o\in M$. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the …
We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained results to slant submanifolds.
In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic …
We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in $N$. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary …
Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliations, foliations on surfaces, and when the EGF is produced by the extrinsic Ricci tensor.
For any hypersurface $M$ of a Riemannian manifold $X$, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvatures. Here we derive explicit formulas for the extrinsic version ${\bf P}_4$ of the Paneitz operator and the corresponding extrinsic fourth-order $Q$-curvature ${\bf Q}_4$ in general dimensions. This result involves a series of obvious local conformal invariants of the embedding $M^4 \hookrightarrow X^5$ (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant $\mathcal{C}$. In tu…
In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $\mathbb{S}^3$ and establish a correspondence between simply connected surfaces having extrinsic curvature $K$, $K$ different from 0 and -1, immersed in $\mathbb{S}^3$ with simply connected surfaces having non-vanishing extrinsic curvature immersed in the euclidean space $\mathbb{…
In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.
We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of the solution procedure is to find (from a system of quasilinear PDE's) the principal curvatures of the foliation. Examples for extrinsic Newton transformation flow, extrinsic Ricci flow, and applications to foliations on surfaces are given.
Extrinsic bi-Conformal Heat Flow and its smoothness
本文提出双共形热流并证明其全局光滑性及无有限时间奇异性。
We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature over all planes that contain vectors from both distributions of an almost-product structure and the variations we consider preserve orthogonality of the distributions. We derive the directional derivative $D J_{\,\rm mix}$ (o…
Piecewise flat approximations of local extrinsic curvature for non-Euclidean embeddings
本文在分段平坦流形上构造平均曲率和有向曲率的离散形式，通过铰链角加权标量和为欧几里得与非欧几里得空间中的光滑流形提供局部曲率近似，并证明了随网格细化收敛于光滑曲率值。
Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotiente $\widetilde{N}^{11}=(Sp(2)\times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotiente $\widetilde{N}^{11}/S^3$, we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere $Σ^7$. Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature…
An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R^3. Of special interest is the Ricci flow on a toroidal surface of revolution, that is, a surface of revolution whose profile curve is an immerse…
We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second order parabolic PDEs, for which the existence/uniquenes and in some cases convergence of a solution are shown. Applications to the problem of prescribing mean curvature function of a codimensio…
On quasicomplete $k$-surfaces in $3$-dimensional space-forms
本文引入拟完备性概念，证明在三维空间形式中常正外在曲率且满足 k>Max(0,-c) 的拟完备浸入曲面必为测地球面，从而完成该类曲面的完整分类。
Let $(X,g)$ be a compact manifold with boundary $M^n$ and $σ$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $σ^{-2}g$. We prove that if $σ$ is an approximate solution of a singular Yamabe problem, these families can be wr…
A mean curvature flow approach to Hamilton's pinching theorem
本文通过均曲率流方法证明了Hamilton的外insic收缩定理，采用几何分析手段解决曲面收缩问题。
Discrete extrinsic curvatures based on polar polyhedra concept
本文利用局部极多面体序列构造球面高斯映射的分片仿射逼近，得到平均曲率和高斯曲率的逐点收敛逼近，并实现弯曲能量的自然离散化。
We prove that smooth convex $α$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\leq CH(x,t)(H(x,t)|x-y|+1)^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.
The Penrose inequality in extrinsic geometry
本文解决了G. Huisken的外蕴几何彭罗斯不等式猜想，证明三维渐近平坦支撑曲面的外质量下界，刻画等号情形的半悬链面结构，并通过极小毛细曲面的非递减量完成证明，为悬链面提供新的几何刻画。
In this paper, we obtain several new intrinsic and extrinsic differential sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $n(\ge 4)$-dimensional Riemannian manifold $M$ is diffeomorphic to $S^n$ if one of the following conditions holds pointwisely: $$ (i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$ Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectional curvature, the $k$-th weak Ricci curvature and the normalized scalar curvature. Fo…
Mass from an Extrinsic Point of View
本文从外蕴角度将浸入子流形的第 $q$ 阶 Gauss–Bonnet–Chern 质量写成总平均曲率与曲率向量—位置向量积分两项的线性组合，并据此导出对应正质量定理成立时的几何不等式。
Spectral Bernstein theorems for submanifolds in Euclidean spaces
本文研究欧几里得空间中子流形的本质谱，探讨了有限总均曲率、外向距离梯度收敛性、外向体积增长或曲率pinching等几何条件，并证明了当子流形的二阶基本形式满足L^p有界时，其本质谱为[0, +∞)。
Complete bounded $λ$-hypersurfaces in the weighted volume-preserving mean curvature flow
本文研究加权保体积平均曲率流中完备有界 λ-超曲面，建立体积比较定理并给出内直径与外半径估计，进而在一般限制条件下刻画其拓扑性质。
For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmor…
We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.
We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.
Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions
该研究针对流形结构数据的线性回归问题，分析了数据流形外在几何（曲率或高阶非线性项）对回归解唯一性的影响，明确了流形几何在保障分布外推理时回归模型稳定性的作用。
Refining Cramér-Rao Bound With Multivariate Parameters: An Extrinsic Geometry Perspective
本文从外在几何视角出发，通过Hilbert空间平方根嵌入推导出修正的Cramér-Rao界，结合第二基本形式和半正定规划，改进了多变量参数的估计性能。
We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.
Modified mean curvature flow of graphs in Riemannian manifolds
本文研究了在具有Killing向量场的Riemannian流形中修改的平均曲率流，得出了先验估计并证明了光滑整体长时间解的存在性。
A Rigidity Result for Minimal Rotation Hypersurfaces of 5D Spaces of Constant Curvature
本文证明五维常曲率空间中极小超曲面上的一个外在逐点不变量非正，且其恒为零当且仅当超曲面为旋转超曲面，并猜想该刚性结果可推广到其他维数。
In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the space of 2 convex hypersurfaces in $\mathbb{R}^{n+1}$.
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