The level set flow of a hypersurface in R4\mathbb R^4 of low entropy does not disconnect

We show that if $\Sigma\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $\lambda(\Sigma)\leq \lambda(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $\Sigma$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.Comment: Strengthened statement of Proposition 3.4 and fixed a gap in proof of Proposition 3.

Low Entropy and the Mean Curvature Flow with Surgery

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $\Lambda_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $H - \frac{\langle x, \nu \rangle}{2} > 0$ is possible and as an application we use the surgery flow to show that smooth $n$-dimensional closed self shrinkers with entropy less than $\Lambda_{n-2}$ are isotopic to the round $n$-sphere.Comment: Revised after referee's report. A number of small errors fixed and minor parts of the argument were altere

Warped Tori with Almost Non-Negative Scalar Curvature

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat $3$-torus.Comment: 21 pages. The second version has no changes to the estimates, just a change in title and some exposition in response to a request by a senior mathematician. Minor revisions made suggested by the referee in version three. To appear in Geometriae Dedicat

Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn

The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in $C^2$, then they converge in $C^{2,\theta}$. As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flo

Quantization of the Energy for the inhomogeneous Allen-Cahn mean curvature

We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ bounds on the Allen--Cahn mean curvature (first variation of the Allen--Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in $L^{q_0}, q_0 > n$. The latter is a diffused version of Allard's convergence theorem for integer rectifiable varifolds.Comment: Acknowledgement update

Some properties of closed hypersurfaces of small entropy and the topology of hypersurfaces through singularities of mean curvature flow

We record in this thesis three results concerning entropy and singularities in mean curvature ow (MCF). The rst result is a stability result of round spheres under small-entropy perturbation. The round spheres are minimizer of the entropy functional and we show that in all dimensions a closed hypersurface must be close to a round sphere in Hausdor distance if the entropy is close to that of a round sphere. This generalizes a result of Bernstein-Wang in dimension 2. The second result gives a sharp entropy lower bound for disconnection to happen in mean curva- ture ow of hypersurfaces in R4. And it’s related to the rst result in that it sharpens the condition of a uniform continuity estimate of Hausdor distance over time. The non-sharp version of this uniform continuity was used as a key lemma in the proof of the rst result. This second result is joint work with J. Benstein. The third result is a rigidity result in the singularity models of mean curvature ow. Self-shrinkers are singularity models in mean curvature ow by Huisken’s monotonicity formula. And by using techniques from minimal surfaces, we showed that a self-shrinking torus must be unknotted. This third result is joint work with A. Mramor

Drive laser system for the DC-SRF photoinjector at Peking University

Photoinjectors are widely used for linear accelerators as electron sources to generate high-brightness electron beam. Drive laser, which determines the timing structure and quality of the electron beam, is a crucial device of photoinjector. A new drive laser system has been designed and constructed for the upgraded 3.5-cell DC-SRF photoinjector at Peking University. The drive laser system consists of a 1064 nm laser oscillator, a four- stage amplifier, the second and fourth harmonic generators, the optical system to transfer the UV pulses to the photocathode, and the synchronization system. The drive laser system has been successfully applied in the stable operation of DC-SRF photoinjector and its performance meets the requirements. 266 nm laser with an average power close to 1W can be delivered to illuminate the Cs2Te photocathode and the instability is less than 5% for long time operation. The design consideration for improving the UV laser quality, a detailed description of laser system, and its performance are presented in this paper.Comment: 6 pages, 8 figures, submit to CP