Jacob Bernstein
This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.
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We introduce a family of functionals on submanifolds of Cartan-Hadamard manifolds that generalize the Colding-Minicozzi entropy of submanifolds of Euclidean space. We show that these functionals are monotone under mean curvature flow under natural conditions. As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon.
Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers
本文利用自收缩子的归一化 Li-Yau 共形体积给出不依赖余维且对平面尖锐的 Colding-Minicozzi 熵下界，并借此验证实射影平面情形的相关猜想。
Rigidity Properties of Colding-Minicozzi Entropies
本文证明了广义 Colding-Minicozzi 熵最小化子的刚性性质，其初等证明适用于熵不单调的情形。
Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $Σ_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $Σ_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.
Minimal surfaces and Colding-Minicozzi entropy in complex hyperbolic space
本文研究复双曲空间中含极小 Lagrangian 子流形的一类极小子流形的渐近正则性，并建立 Colding-Minicozzi 熵与由渐近几何定义的 CR-体积之间的联系。
Colding-Minicozzi entropies of some self-shrinkers
本文数值估计了某些自收缩子的Colding-Minicozzi熵，发现n维Angenent环面的熵随维数n递减，部分回答了Berchenko-Kogan的问题。
We consider operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $L = Δ+ V +a K.$ Here $Δ$ is the Laplacian of $Σ$, $V$ a non-negative potential on $Σ$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $Σ$ immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume L is nonpositive acting on functions compactly supported on $Σ$ and we obtain results in the spirit of some theorems of Ficher-Colbrie-Schoen, Colding-Minicozzi, and Castillon. We extend th…
Following work of Colding-Minicozzi, we define a notion of entropy for connections over $\mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying "generic singularities" of Yang-Mills flow, and we discuss the differences in this strate…
We prove Łojasiewicz inequalities for round cylinders and cylinders over Abresch-Langer curves, using perturbative analysis of a quantity introduced by Colding-Minicozzi. A feature is that this auxiliary quantity allows us to work essentially at first order. This new method interpolates between the higher order perturbative analysis used by the author for certain shrinking cylinders, and the differential geometric method used by Colding-Minicozzi for the round case.
Colding and Minicozzi have shown that an embedded minimal disk $0\inΣ\subset B_R$ in $\Real^3$ with large curvature at 0 looks like a helicoid on the scale of $R$. Near 0, this can be sharpened: on the scale of $|A|^{-1}(0)$, $Σ$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by Colding and Minicozzi to see this description cannot hold on the scale $R$.
We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature.
We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews. In the proof, we use the concept of the width of a hypersurface, introduced by Colding and Minicozzi. We also extend the result to 2-convex hypersurfaces, using the 2-width.
Self-shrinkers are the special solutions of mean curvature flow in $\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow. In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White, namely that any closed hypersurface in $\mathbf{R}^{n+1}$ has entropy at least that of the round sphere, holds in any dimension $n$. This result had previously been e…
Motivated and inspired by the recent work of Colding [5] and Colding-Minicozzi [6] we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in [5, 6].
Cylindrical tangent flows in mean curvature flow
本文针对平均曲率流的柱形奇点，采用受Székelyhidi启发的新方法，证明其唯一性与刚性结果，完善了Colding-Minicozzi相关研究的结论
Using the lamination theory developed by Colding and Minicozzi for sequences of embedded, finite genus minimal surfaces with boundaries going to infinity \cite{CM5}, we show that the space of genus-one helicoids is compact (modulo rigid motions and homotheties). This generalizes a result of Hoffman and White \cite{HW}.
In this paper, we first use the method of Colding and Minicozzi [5] to show that K. Smoczyk's classification theorem [16] for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.
Integral of scalar curvature on non-parabolic manifolds
本研究借助Colding与Minicozzi的单调性公式，证明了带非负Ricci曲率的三维完备非抛物黎曼流形上，标量曲率的渐近加权缩放不变积分存在由渐近体积比表示的显式界。
On self shrinkers of medium entropy in $\mathbb{R}^4$
本文研究四维欧氏空间中熵不超过 $Λ_1$ 的光滑渐近锥形自收缩子，推进中等熵自收缩子的分类理解。
In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into $\R^\ell$. We are also lead to study the continuity of the entropy functional in certain cases.
In this paper, we generalize Colding and Minicozzi's work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the $F$-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in \cite{An} are unstable.
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