arXiv research

Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.

Entropy derived from Colding's volume on Ricci-flat manifolds.

Study minimal surfaces in complex hyperbolic space, linking entropy and volume.

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.

Entropy for submanifolds in hyperbolic space defined.

New proof linking scalar curvature to volume growth on 3-manifolds.

We use the Dong-Sogge-Zelditch formula to obtain a lower bound for the volume of the nodal sets of eigenfunctions. Our result improves the recent results of Sogge-Zelditch and in dimensions n \leq 5 gives a new proof for the lower bounds of Colding-Minicozzi.

Proves effective linear volume growth for 3-manifolds with positive scalar curvature.

Formula connects curvature to volume in special geometric spaces.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Let $M$ be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the p…

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compac…

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold $(M^3, g)$ with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

Kahler manifolds with specific curvature properties are close to projective spaces.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…

NetDP predicts loan defaults using network data, addressing cold-start issues.

We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] on manifolds to graphs: For a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dime…

Sharp gradient estimates for positive Ricci curvature manifolds.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $\ge n-1$. In \cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)\to 0$; 2) the volume of $M$ ${\text{Vol}}(M)\to{\text{Vol}}(S^n)$; 3) the radius of $M…

Let $(M,\bar{g}, e^{-f}dμ)$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compa…

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

Study on cold and freezing sets in digital images.

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the c…

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the …

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to that of the unit n sphere is close (in the Gromov-Hausdorff distance) to the sphere…

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

Numerically estimates Colding-Minicozzi entropies of self-shrinkers.

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p in M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff…

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians:…

SPARC tackles cold-start nodes in graphs by using spectral embeddings.

It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations. The special case $m=3$ proves the Double Bubble problem for the Gaussian measure,…

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close…

Bayesian neural networks with data augmentation show a persistent cold posterior effect.

Playlist recommendation involves producing a set of songs that a user might enjoy. We investigate this problem in three cold-start scenarios: (i) cold playlists, where we recommend songs to form new personalised playlists for an existing user; (ii) cold users, where we recommend songs to form new playlists for a new us…

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

Graph neural networks improve cold start for new items in recommender systems.

The item cold-start problem seriously limits the recommendation performance of Collaborative Filtering (CF) methods when new items have either none or very little interactions. To solve this issue, many modern Internet applications propose to predict a new item's interaction from the possessing contents. However, it is…

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

Study shows rigidity for entropy minimizers in non-monotone cases.

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…

Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the actio…

The cold posterior effect is explored through PAC-Bayes bounds for small sample sizes.

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

Cold posteriors in BNNs harm performance, likely due to incorrect likelihood.