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Autoregressive next-token prediction is a standard pretraining method for large-scale language models, but its application to vision tasks is hindered by the non-sequential nature of image data, leading to cumulative errors. Most vision models employ masked autoencoder (MAE) based pretraining, which faces scalability issues. To address these challenges, we introduce \textbf{TokenUnify}, a novel pretraining method that integrates random token prediction, next-token prediction, and next-all token prediction. We provide theoretical evidence demonstrating that TokenUnify mitigates cumulative errors in visual autoregression. Cooperated with TokenUnify, we have assembled a large-scale electron microscopy (EM) image dataset with ultra-high resolution, ideal for creating spatially correlated long sequences. This dataset includes over 120 million annotated voxels, making it the largest neuron segmentation dataset to date and providing a unified benchmark for experimental validation. Leveraging the Mamba network inherently suited for long-sequence modeling on this dataset, TokenUnify not only reduces the computational complexity but also leads to a significant 45\% improvement in segmentation performance on downstream EM neuron segmentation tasks compared to existing methods. Furthermore, TokenUnify demonstrates superior scalability over MAE and traditional autoregressive methods, effectively bridging the gap between pretraining strategies for language and vision models. Code is available at \url{https://github.com/ydchen0806/TokenUnify}.

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincar\'e-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincar\'e-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincar\'e-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.

The electron microscope (EM) remains the predominant technique for elucidating intricate details of the animal nervous system at the nanometer scale. However, accurately reconstructing the complex morphology of axons and myelin sheaths poses a significant challenge. Furthermore, the absence of publicly available, large-scale EM datasets encompassing complete cross sections of the corpus callosum, with dense ground truth segmentation for axons and myelin sheaths, hinders the advancement and evaluation of holistic corpus callosum reconstructions. To surmount these obstacles, we introduce the AxonCallosumEM dataset, comprising a 1.83 times 5.76mm EM image captured from the corpus callosum of the Rett Syndrome (RTT) mouse model, which entail extensive axon bundles. We meticulously proofread over 600,000 patches at a resolution of 1024 times 1024, thus providing a comprehensive ground truth for myelinated axons and myelin sheaths. Additionally, we extensively annotated three distinct regions within the dataset for the purposes of training, testing, and validation. Utilizing this dataset, we develop a fine-tuning methodology that adapts Segment Anything Model (SAM) to EM images segmentation tasks, called EM-SAM, enabling outperforms other state-of-the-art methods. Furthermore, we present the evaluation results of EM-SAM as a baseline., Comment: 9 pages, 6 figures

We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitational instantons and ALG gravitational instantons which are of order $2$. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathscr{P}$, which we show is surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct some new degenerations of hyperk\"ahler metrics on the K3 surface which exhibit bubbling of ALG$^*$ gravitational instantons.

We show that any asymptotically Calabi manifold which is Calabi-Yau can be compactified complex analytically to a weak Fano manifold. Furthermore, the Calabi-Yau structure arises from a generalized Tian-Yau construction on the compactification, and we prove a strong uniqueness theorem. We also give an application of this result to the surface case.

We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on ALG$^*$ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG$^*$ manifolds with non-negative Ricci curvature having group $\Gamma = \{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of the first betti number for any ALG$^*$ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG$^*$ gravitational instantons., Comment: 35 pages; final version; to appear in J. Reine Angew. Math. (Crelle's Journal)

We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special K\"ahler metric on the 2-sphere, or the unit interval. (2) Any complete hyperk\"ahler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity., Comment: A conjecture on the L2-curvature energy was added in Section 7.3. To appear in Acta Mathematica

In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity., Comment: Minor updates. The post [arXiv:1906.03368v1] has been split into two papers. This one is essentially extracted from Section 5 and the Appendix in [arXiv:1906.03368v1]. To appear in Calculus of Variations and Partial Differential Equations

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers., Comment: 77 pages

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way, we obtain a complete result in all dimensions establishing explicit and precise relationships between the metric collapsing and complex structure degenerations. This result is new even in complex dimension two. This is achieved via gluing and singular perturbation techniques, and a key geometric ingredient involving the construction of certain (not necessarily smooth) K\"ahler metrics with torus symmetry. We also discuss possible extensions of this result to more general settings., Comment: Minor revision and updates

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds., Comment: 95 pages

In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension $<\frac{n-2}{2}$. If additionally we assume that the non-local curvature $Q_{2\gamma}\geq 0$ for some $1<\gamma<2$, the Hausdorff dimension of the limit set is less than or equal to $\frac{n-2\gamma}{2}$. If $Q_{2\gamma}>0$, then the above inequality is strict. Moreover, the above upper bound is sharp. As applications, we obtain some topological rigidity and classification theorems in lower dimensions., Comment: We added some new rigidity results for nonnegative fractional curvature (Theorem 1.3, Theorem 1.4, Theorem 1.5). Also we obtained a sharp upper bound in the nonnegative fractional curvature case (see Theorem 1.1)

In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian $(-\triangle)^{\alpha/2}$ for $0<\alpha<2$, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify the non-negative solutions for a semilinear equation involving the fractional Laplacian in $\mathbb{R}^n$; we prove a non-existence result for prescribing $Q_{\alpha}$ curvature equation on $\mathbb{S}^n$; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on the half Euclidean space. We expect to see more applications of this method to many other equations involving non-local operators.

In this paper we discuss and prove $\epsilon$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset \mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ for $\ell\leq 3$, then in fact $|Rm|\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy to see without more such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on $B_1(x)$. More precisely, using established techniques one can see there exists $\epsilon(n)$ such that if $(M^n,g)$ is an Einstein manifold and $B_2(x)$ is $\epsilon$-Gromov-Hausdorff close to ball in $B_2(0^{k-\ell},z^*)\subset\mathbb{R}^{k-\ell}\times Z^\ell$, then the fibered fundamental group $\Gamma_\epsilon(x)\equiv Image[\pi_1(B_\epsilon(x))\to\pi_1(B_2(x))]$ is almost nilpotent with $rank(\Gamma_{\epsilon}(x))\leq n-k$. The main result of the this paper states that if $rank(\Gamma_{\epsilon}(x))= n-k$ is maximal, then $|Rm|\leq C$ on $B_1(x)$. In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.