Numerical Ricci-flat metrics on K3

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale of days using a desktop computer. We compute various geometric and spectral quantities from our numerical metrics. Using similar resources we expect our methods to practically extend to Calabi-Yau three-folds with a high degree of discrete symmetry, although we expect the general three-fold to remain a challenge due to memory requirements.Comment: 38 pages, 10 figures; program code and animations of figures downloadable from http://schwinger.harvard.edu/~wiseman/K3/ ; v2 minor corrections, references adde

Extrinsic Ricci Flow on Surfaces of Revolution

An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in Rn+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 R3. Our approach is to build the desired embedded surfaces of revolution S(t) in R3 into the flow at the outset by rewriting the Ricci flow equations in terms of extrinsic geometric quantities in a natural way. This identifies an extrinsic representation with a particular solution of the scalar logarithmic diffusion equation in one space variable. The result is a single, unified framework to construct an extrinsic representation in R3 of a Ricci flow on a surface of revolution S initialized by a metric g0. Of special interest is the Ricci flow on the torus S1 ×S1 embedded in R3. In this case, the extrinsic representation of the Ricci flow on a Riemannian cover of S is eternal. This flow can also be realized as a compact family of nonsmooth, but isometric, embeddings of the torus into R3

Representation Learning via Manifold Flattening and Reconstruction

This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening Networks (FlatNet), are theoretically interpretable, computationally feasible at scale, and generalize well to test data, a balance not typically found in manifold-based learning methods. We present empirical results and comparisons to other models on synthetic high-dimensional manifold data and 2D image data. Our code is publicly available.Comment: 44 pages, 19 figure