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We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-White, we show that any level set flow starting from an embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, we obtain that the problem of evolving embedded 2-spheres via the mean curvature flow equation is well-posed within a natural class of singular solutions. Second, we use our result to remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows starting from any generic embedded surface only incur cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities. Among other things, this theory also applies to the innermost and outermost flow of any embedded surface and shows that all singularity models of such flows must have multiplicity one. It also establishes equality of the fattening time with the discrepancy time. Lastly, we obtain a number of further results characterizing a separation phenomenon of mean curvature flows of surfaces., Comment: 58 pages. Enhanced the regularity statement (Theorem 1.9). Added proof of Ilmanen's finite topology conjecture (Theorem 1.10). Made a number of minor changes and corrections

Multi-baseline interferometric synthetic aperture radar (InSAR) techniques are effective approaches for retrieving the 3-D information of urban areas. In order to obtain a plausible reconstruction, it is necessary to use large-stack interferograms. Hence, these methods are commonly not appropriate for large-scale 3-D urban mapping using TanDEM-X data where only a few acquisitions are available in average for each city. This work proposes a new SAR tomographic processing framework to work with those extremely small stacks, which integrates the non-local filtering into SAR tomography inversion. The applicability of the algorithm is demonstrated using a TanDEM-X multi-baseline stack with 5 bistatic interferograms over the whole city of Munich, Germany. Systematic comparison of our result with airborne LiDAR data shows that the relative height accuracy of two third buildings is within two meters, which outperforms the TanDEM-X raw DEM. The promising performance of the proposed algorithm paved the first step towards high quality large-scale 3-D urban mapping., Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:2003.07803

We develop a new degree theory for 4-dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over $S^3$ with non-negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to $S^3/\Gamma$ if we allow the expanding soliton to have orbifold singularities. Our theory reveals the existence of a new topological invariant, called the expander degree, applicable to a particular class of compact, smooth 4-orbifolds with boundary. This invariant is roughly equal to a signed count of all possible gradient expanding solitons that can be defined on the interior of the orbifold and are asymptotic to any fixed cone metric with non-negative scalar curvature. If the expander degree of an orbifold is non-zero, then gradient expanding solitons exist for any such cone metric. We show that the expander degree of the 4-disk $D^4$ and any orbifold of the form $D^4/\Gamma$ equals 1. Additionally, we demonstrate that the expander degree of certain orbifolds, including exotic 4-disks, vanishes. Our theory also sheds light on the relation between gradient and non-gradient expanding solitons with respect to their asymptotic model. More specifically, we show that among the set of asymptotically conical expanding solitons, the subset of those solitons that are gradient forms a union of connected components., Comment: 121 pages, minor updates

We prove the existence of a unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two complex dimensions., Comment: 11 pages

We show that every closed, oriented, topologically PSC 4-manifold can be obtained via 0 and 1-surgeries from a topologically PSC 4-orbifold with vanishing first Betti number and second Betti number at most as large as the original one., Comment: 15 pages. Exposition improved and example given to explain the necessity of orbifolds. To appear in Adv. Math

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than $r^{\frac{n+1}{2}}.$ This result not only improves the estimate in [CMZ21b, Theorem 1.3], but also is optimal since the Bryant soliton and Appleton's solitons [Ap17] have exactly this growth rate., Comment: 11 pages

We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $\kappa$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity., Comment: 9 pages

Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $\text{Nil}$. We show that the diffeomorphism group $\text{Diff}(X)$ deformation retracts to the isometry group $\text{Isom}(X)$. Combining this with earlier work by many authors, this completes the determination the homotopy type of $\text{Diff}(X)$ for any compact, orientable, prime 3-manifold $X$., Comment: 14 pages, minor updates

Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.

This is a survey on recent developments in Ricci flows., Comment: 16 pages

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $\mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general., Comment: 13 pages

In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension $\geq 4$. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelman's results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications., Comment: 155 pages; Comments and questions welcome; improved introduction and minor edits

We develop a compactness theory for super Ricci flows, which lays the foundations for the partial regularity theory in [Bam20b]. Our results imply that any sequence of super Ricci flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow. We will study the geometric and analytic properties of this limiting flow, as well as the convergence in detail. We will also see that, under appropriate local curvature bounds, a limit of Ricci flows can be decomposed into a regular and singular part. The regular part can be endowed with a canonical structure of a Ricci flow spacetime and we have smooth convergence on a certain subset of the regular part., Comment: 117 pages; minor corrections; Comments and questions welcome

In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an $L^1$-Poincar\'e inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local $\varepsilon$-regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows., Comment: 48 pages; Comments and questions welcome; updated introduction and minor edits

This paper addresses the general problem of single-look multi-master SAR tomography. For this purpose, we establish the single-look multi-master data model, analyze its implications for single and double scatterers, and propose a generic inversion framework. The core of this framework is nonconvex sparse recovery, for which we develop two algorithms: one extends the conventional nonlinear least squares (NLS) to the single-look multi-master data model, and the other is based on bi-convex relaxation and alternating minimization (BiCRAM). We provide two theorems for the objective function of the NLS subproblem, which lead to its analytic solution up to a constant phase angle in the one-dimensional case. We also report our findings from the experiments on different acceleration techniques for BiCRAM. The proposed algorithms are applied to a real TerraSAR-X data set, and validated with height ground truth made available via a SAR imaging geodesy and simulation framework. This shows empirically that the \emph{single-master} approach, if applied to a single-look \emph{multi-master} stack, can be insufficient for layover separation, and the \emph{multi-master} approach can indeed perform slightly better (despite being computationally more expensive) even in the case of single scatterers. Besides, this paper also sheds light on the special case of single-look bistatic SAR tomography, which is relevant for current and future SAR missions such as TanDEM-X and Tandem-L.

Deep learning in remote sensing has become an international hype, but it is mostly limited to the evaluation of optical data. Although deep learning has been introduced in Synthetic Aperture Radar (SAR) data processing, despite successful first attempts, its huge potential remains locked. In this paper, we provide an introduction to the most relevant deep learning models and concepts, point out possible pitfalls by analyzing special characteristics of SAR data, review the state-of-the-art of deep learning applied to SAR in depth, summarize available benchmarks, and recommend some important future research directions. With this effort, we hope to stimulate more research in this interesting yet under-exploited research field and to pave the way for use of deep learning in big SAR data processing workflows., Comment: article accepted by IEEE Geoscience and Remote Sensing Magazine. Copyright may be transferred without notice, after which this version may no longer be accessible

Multi-baseline interferometric synthetic aperture radar (InSAR) techniques are effective approaches for retrieving the 3-D information of urban areas. In order to obtain a plausible reconstruction, it is necessary to use more than twenty interferograms. Hence, these methods are commonly not appropriate for large-scale 3-D urban mapping using TanDEM-X data where only a few acquisitions are available in average for each city. This work proposes a new SAR tomographic processing framework to work with those extremely small stacks, which integrates the non-local filtering into SAR tomography inversion. The applicability of the algorithm is demonstrated using a TanDEM-X multi-baseline stack with 5 bistatic interferograms over the whole city of Munich, Germany. Systematic comparison of our result with TanDEM-X raw digital elevation models (DEM) and airborne LiDAR data shows that the relative height accuracy of two third buildings is within two meters, which outperforms the TanDEM-X raw DEM. The promising performance of the proposed algorithm paved the first step towards high quality large-scale 3-D urban mapping.

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case., Comment: 111 pages

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $\kappa$-solutions. In this paper, we present an alternative proof for this fact and show that compact $\kappa$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows., Comment: 20 pages

Tomographic SAR (TomoSAR) inversion of urban areas is an inherently sparse reconstruction problem and, hence, can be solved using compressive sensing (CS) algorithms. This paper proposes solutions for two notorious problems in this field: 1) TomoSAR requires a high number of data sets, which makes the technique expensive. However, it can be shown that the number of acquisitions and the signal-to-noise ratio (SNR) can be traded off against each other, because it is asymptotically only the product of the number of acquisitions and SNR that determines the reconstruction quality. We propose to increase SNR by integrating non-local estimation into the inversion and show that a reasonable reconstruction of buildings from only seven interferograms is feasible. 2) CS-based inversion is computationally expensive and therefore barely suitable for large-scale applications. We introduce a new fast and accurate algorithm for solving the non-local L1-L2-minimization problem, central to CS-based reconstruction algorithms. The applicability of the algorithm is demonstrated using simulated data and TerraSAR-X high-resolution spotlight images over an area in Munich, Germany., Comment: 10 pages

This paper presents a nonlocal InSAR filter with the goal of generating digital elevation models of higher resolution and accuracy from bistatic TanDEM-X strip map interferograms than with the processing chain used in production. The currently employed boxcar multilooking filter naturally decreases the resolution and has inherent limitations on what level of noise reduction can be achieved. The proposed filter is specifically designed to account for the inherent diversity of natural terrain by setting several filtering parameters adaptively. In particular, it considers the local fringe frequency and scene heterogeneity, ensuring proper denoising of interferograms with considerable underlying topography as well as urban areas. A comparison using synthetic and TanDEM-X bistatic strip map datasets with existing InSAR filters shows the effectiveness of the proposed techniques, most of which could readily be integrated into existing nonlocal filters. The resulting digital elevation models outclass the ones produced with the existing global TanDEM-X DEM processing chain by effectively increasing the resolution from 12m to 6m and lowering the noise level by roughly a factor of two., Comment: Paper has been accepted to be published in IEEE Transaction on Geoscience and Remote Sensing

$L_1$ regularization is used for finding sparse solutions to an underdetermined linear system. As sparse signals are widely expected in remote sensing, this type of regularization scheme and its extensions have been widely employed in many remote sensing problems, such as image fusion, target detection, image super-resolution, and others and have led to promising results. However, solving such sparse reconstruction problems is computationally expensive and has limitations in its practical use. In this paper, we proposed a novel efficient algorithm for solving the complex-valued $L_1$ regularized least squares problem. Taking the high-dimensional tomographic synthetic aperture radar (TomoSAR) as a practical example, we carried out extensive experiments, both with simulation data and real data, to demonstrate that the proposed approach can retain the accuracy of second order methods while dramatically speeding up the processing by one or two orders. Although we have chosen TomoSAR as the example, the proposed method can be generally applied to any spectral estimation problems., Comment: 11 pages, IEEE Transactions on Geoscience and Remote Sensing

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature., Comment: 29 pages, 1 figure

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper., Comment: 182 pages, 10 figures, minor corrections

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds)., Comment: References added, changes in Cor 4 and in Remarks after

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $\geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows., Comment: 71 pages, formerly part of arXiv:1512.08527

In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded curvature, which we will describe in a subsequent paper., Comment: 105 pages, minor updates, formerly part of arXiv:1512.08527

In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away from a set of codimension $\geq 4$. The result has two main consequences: First, it implies that singularities in Ricci flows with bounded scalar curvature have codimension $\geq 4$ and, second, it establishes a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. In the course of the proof, we will also establish the following results: $L^{p < 4}$ curvature bounds, integral bounds on the curvature radius, Gromov-Hausdorff closeness of time-slices, an $\varepsilon$-regularity theorem for Ricci flows and an improved backwards pseudolocality theorem., Comment: This article has been divided into the following two articles: "Structure theory of singular spaces" (arXiv:1603.05236) and "Convergence of Ricci flows with bounded scalar curvature" (arXiv:1603.05235)

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.

In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $\frac12$-H\"older continuous in a uniform sense. This implies that the distance function can be extended continuously up to the singular time., Comment: 13 pages

In this note we reprove a theorem of Gromov using Ricci flow. The theorem states that a, possibly non-constant, lower bound on the scalar curvature is stable under $C^0$-convergence of the metric., Comment: 9 pages

In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform $L^2$ curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger $\varepsilon$-regularity theorem for Ricci flows. This result is particularly useful in the study of K\"ahler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results., Comment: 42 pages, minor updates

In the following series of papers we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by $C t^{-1}$. This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as $t \to \infty$., Comment: 16 pages, 3 figures

This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman's techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed., Comment: 58 pages, revised version

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. We remark that in this paper we will mostly be dealing with non-singular Ricci flows. The existence of surgeries will not play an important role., Comment: 40 pages, revised version

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used., Comment: 47 pages, revised version

This is the fourth and last part of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes non-singular eventually and the curvature is bounded by $C t^{-1}$. The second result provides a qualitative description of the geometry as $t \to \infty$., Comment: 100 pages, 15 figures, revised version