Jose Torres Santaella
The main result in this paper is a non-existence Theorem of entire $Q_{n-1}$-translators in $\mathbb{R}^{n+1}$. In addition, an example of non-entire complete $Q_{n-1}$-translator has been found and a Tangential Principle for $Q_k$-translators in $\mathbb{R}^{n+1}$.
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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…
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。
Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe
该研究针对双曲爱因斯坦静态宇宙中平均曲率流的非退化平移子，分类了水平球面膜状与旋转不变的类空、类时平移子，证明了类空平移子的唯一性与等距延拓定理，并给出边界为球和矩形时的平移子刻画。
In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n \geq 2$ in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely that the profile curve is convex, we remove the entropy assumption. Supposing the profile curve is additionally reflection symmetric we show there are only finitely many such shrinkers up to rigid motion.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.
The Classification of Rotationally symmetric hypersurfaces in the Heisenberg groups $H_{n}$
该研究针对海森堡群$H_n$中的旋转对称超曲面，证明基本定理并结合已有结果完成脐型超曲面分类，刻画常$p$-平均曲率旋转对称超曲面的生成曲线，验证Alexandrov定理对这类超曲面的有效性。
On March's criterion for transience on rotationally symmetric manifolds
本文将弱模型上有界非常值调和函数存在性的March判据推广到旋转对称度量的轻微推广情形，证明其等价于无穷远处Dirichlet问题的可解性。
Rotationally symmetric Ricci Flow on $\mathbb{R}^{n+1}$
本文在尺度不变曲率界下建立了 $\mathbb{R}^{n+1}$ 上旋转对称度量的完整 Ricci 流短时存在理论，并构造出源于原点含锥状奇点度量的完整 Ricci 流解。
In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric $n$-dimensional hypersurface in $\R^{n+1}$ between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.
In this paper we study a boundary value problem for the Ricci flow in the two dimensional ball endowed with a rotationally symmetric metric. We show short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally we show that if the initial metric has positive Gaussian curvature and the boundary positive geodesic curvature then the flow uniformizes de curvature along a sequence of times.
In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry. Some applications to isoperimetric problems are also given.
Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost ever…
Existence proofs for rotationally symmetric translating solutions to mean curvature flow
本文针对平均曲率流旋转对称平移解，提出基于奇异常微分方程的新存在性证明方法，避免使用偏微分方程技术。
Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
本文在旋转对称 Cartan-Hadamard 流形上引入全局频率函数，并在实双曲空间中证明相关频率函数的单调性，进而得到对数凸性定理和双曲三球定理。
Given a smooth, symmetric, homogeneous of degree one function $f\left(λ_{1},\cdots,\,λ_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\,n$, and a rotationally symmetric cone $\mathcal{C}$ in $\mathbb{R}^{n+1}$, we show that there is a $f$ self-shrinker (i.e. a hypersurface $Σ$ in $\mathbb{R}^{n+1}$ which satisfies $f\left(κ_{1},\cdots,\,κ_{n}\right)+\frac{1}{2}X\cdot N=0$, where $X$ is the position vector, $N$ is the unit normal vector, and $κ_{1},\cdots,\,κ_{n}$ are principal curvatures of $Σ$) that is asymptotic to $\mathcal{C}$ at infinity.
Morse index and nullity of rotationally symmetric $Y$-singular minimal surfaces
本文在Y-奇异性构成单个圆周的假设下，通过固定边界问题和Dirichlet-to-Neumann映射计算旋转对称极小Y-奇异曲面的Morse指标，并证明Y-悬链面的指标为1。
The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that $m=4$ and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is ${\mathbb R}^4$. In the second part of t…
We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.
We study the Ricci flow on $\mathbb{R}^{n+1}$, with $n\geq 2$, starting at some complete bounded curvature rotationally symmetric metric $g_{0}$. We first focus on the case where $(\mathbb{R}^{n+1},g_{0})$ does not contain minimal hyperspheres; we prove that if $g_{0}$ is asymptotic to a cylinder then the solution develops a Type-II singularity and converges to the Bryant soliton, while if the curvature of $g_{0}$ decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotica…
In this paper, we study rotationally symmetric extremal Kähler metrics on $\mathbb C^n$ ($n\geq 2$) and $\mathbb C^2 \backslash\{0\}$. We present a classification of such metrics based on the zeros of the polynomial appearing in Calabi's Extremal Equation. As applications, we prove that there are no $U(n)$ invariant complete extremal Kähler metrics on $\mathbb C^n$ with positive bisectional curvature, and we give a smooth extension lemma for $U(n)$ invariant extremal Kähler metrics on $\mathbb{C}^n\backslash\{0\}$. We retrieve known examples of smooth or singular extremal Kähler metrics on Hir…
On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$
本文对 $\mathbb H^2\times\mathbb R$ 中完备平移子的渐近边界进行分类，证明竖直边界分量为点乘区间或直线，水平边界分量为完备测地线，方法基于对称平移子屏障。
We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.
We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investigated.
Existence of rotationally symmetric embedded f-minimal tori
该研究推广了Angenent的收缩环面，证明了特定欧氏空间中满足条件的旋转对称嵌入f-极小环面的存在性。
The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces
该文研究静态旋转对称空间中的局部约束曲率流，证明其图状解的长时间存在性与指数收敛性、近切片初始超曲面的静态凸保持性，并据此推导一类静态凸域的加权几何不等式。
We show that any smooth solution to the mean curvature flow equations coming out of a rotationally symmetric double cone is also rotationally symmetric.
A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms o…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of rotationally symmetric maps between 2-spheres. We also find many examples of proper biharmonic maps defined locally on a 2-sphere. Our results seem to suggest that any biharmonic map $S^2\longrightarrow (N^n, h)$ be a weakly conformal immersion.
We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.
In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular that the curvature decreases at the same rate at which it blew up. This is the first example of Ricci flow starting from a type-II singularity.
Width Stability of Rotationally Symmetric Metrics
本文证明了旋转对称条件下三维球面极小极大宽度稳定性的Marques-Neves猜想，并将其推广至任意维度$n\geqslant 3$。
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove o Asymptotics of counting function of resonances at large radius o Inverse problem: The rotation radius is uniquely determined by its eigenvalues and resonances. Moreover, there exists an algorithm to recover the rotation radius from its eigenvalues…
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole embedded in the Euclidean space R4. At the end of this paper, we also show the advantage and independence of our main result comparing with those existent results for higher dimensional quantum layers or quantum tubes.
Essential $p$-capacity-volume estimates for rotationally symmetric manifolds
本文针对旋转对称流形上的相对 $p$-容量建立基本体积估计，并用于推导尖锐弱 $(p,q)$-嵌入及主 $p$-频率的精确下界。
We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance'' between a rotationally symmetric manifold and the Euclidian space, which are stated in terms of the ADM mass of the manifold. Importantly, we make explicit the dependencies and scales involved in this problem, particularly the ADM mass, the depth, and the CMC reference hypersu…
Scalar curvature comparison of rotationally symmetric sets
本文受Gromov启发，通过平均曲率、诱导度量和二面角比较，证明旋转对称弱凸参考模型下满足边界条件的非负标量曲率3维流形必平坦，并推广至双曲及$\mathbb{S}^1$对称情形。
Maximum Principles and Consequences for $γ$-translators in $\mathbb{R}^{n+1}$
本文利用极大值原理研究一般外蕴曲率流中的 $γ$-平移子，证明切向原理、闭解与整解不存在性，并确立渐近于圆柱的严格凸单端解必为“碗”型解。
In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a gra…
We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its $Γ$-limit. In particular, we prove that limit vesicles are $C^1$ across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and numerical studies. Moreover, a limit membrane may consist of several topological spheres, which are connected at the axis of revolution and resemble complete buds of the vesicle.
Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers
通过分析旋转对称自收缩子的轮廓曲线，利用Sturm型比较定理证明水平点轨迹的单调性，为Angenent环面的唯一性问题提供新方法。
In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.
We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding. We study it from the point of view of bihamiltonian geometry. We show how to algebraically integrate it by means of the method of separation of variables.
We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity.
This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the …
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold…
Rotationally Symmetric Zoll Metrics with a Cubic Integral
本文提出一种构造Zoll度量的新方法，利用Funk形式与代数关系，给出覆盖所有旋转对称Zoll度量的双参数族，并与早期结果比较。
Symmetry Breaking and Moduli for Ancient Yamabe Flows on the Sphere
对于每个n≥3，构造了单位球面上不可数多个II型古Yamabe流，它们互不共形等价且非旋转对称，具有不定Ricci曲率，后向极限为有限个球面的楔和。
Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class
本文研究了在流形共形类中拉普拉斯特征值的临界度量，通过构造旋转对称调和映射，改进了关于最大特征值的已有结果，并给出了Otsuki tori的显式参数化。
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces
研究了旋转对称超曲面的逆均值流，证明其收敛性并扩展了Minkowski不等式，同时建立了与极小曲面理论的联系。
In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\R^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\times \R^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist non-rotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire con…
Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
本文通过不动点论证，证明了高维空间中具有特定度量形式的奇异旋转对称稳态和膨胀梯度Ricci孤立子的存在性，给出了方程解的无穷多性、渐近行为及唯一全局奇异解的存在条件。
A new proof for the existence of rotationally symmetric gradient Ricci solitons
本文通过求解并分析一个径向 ODE，给出 n+1 维旋转对称稳态和膨胀梯度 Ricci 孤立子存在唯一性的新证明，并刻画解的解析性与渐近行为。
In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that f…
When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to $ -\frac{1}{r}$, where $r$ stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rotationally symmetric one and derive growth estimates of solutions to the eigenvalue equation, from which the absence of eigenvalues will follows.
We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the Type II singularity is accurately modeled by the rotationally symmetric translating soliton.
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