arXiv subjects

Chanyoung Sung

Publications and source records attributed to Chanyoung Sung.

At least 19 recordsLinked to original sources

arXiv2026

Geometric separation and constructive universal approximation with two hidden layers

We give a geometric construction of neural networks that separate disjoint compact subsets of $\Bbb R^n$, and use it to obtain a constructive universal approximation theorem. Specifically, we show that networks with two hidden layers and either a sigmoidal activation (i.e., strictly monotone bounded continuous) or the ReLU activation can approximate any real-valued continuous function on an arbitrary compact set $K\subset\Bbb R^n$ to any prescribed accuracy in the uniform norm. For finite $K$, the construction simplifies and yields a sharp depth-2 (single hidden layer) approximation result.

cs.LG↗

arXiv2025

On the eigenvalues of the Laplacian on fibred manifolds

We prove various comparison theorems of the $i$-th eigenvalue $λ_i$ of the Laplacian on fibred Riemannian manifolds by using fiberwise spherical and Euclidean (or hyperbolic) symmetrization. In particular we generalize the Lichnerowicz inequality and the Faber-Krahn inequality to fiber bundles, and prove a counterpart to Cheng's $λ_1$ comparison theorem under a lower Ricci curvature bound. By applying these, it is shown that $λ_1,\cdots,λ_k$ of a fiber bundle given by a Riemannian submersion with totally geodesic fibers of sufficiently positive Ricci curvature are respectively equal to $λ_1,\cdots,λ_k$ of its base, and $λ_1$ of a (possibly singular) fibration with Euclidean subsets as fibers is no less than $λ_1$ of the disk bundle obtained by replacing each fiber with a Euclidean disk of the same dimension and volume.

math.DG↗

arXiv2025

On the Yamabe constants of product manifolds

By using the fiberwise spherical symmetrization we give a comparison theorem of Yamabe constants on warped products and prove the existence of radially-symmetric Yamabe minimizers on Riemannian manifolds given by products with round spheres.

math.DG↗

arXiv2024

CR Yamabe constant and inequivalent CR structures

The CR Yamabe constant is an invariant of a compact strongly pseudoconvex CR manifold and plays an important role in CR geometry. We show some integral formulae of the CR Yamabe constant. We also construct an infinite-dimensional family of strongly pseudoconvex CR structures with varying CR Yamabe constants and a compact simply-connected manifold admitting two strongly pseudoconvex CR structures with different signs of the CR Yamabe constant.

math.DG↗

arXiv2023

Fiberwise symmetrizations for variational problems on fibred manifolds

We establish a framework for fiberwise symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to prove a comparison theorem of the first eigenvalue of the Laplacian on a warped product manifold.

math.DG↗

arXiv2023

Kummer-type constructions of almost Ricci-flat 5-manifolds

A smooth closed manifold $M$ is called almost Ricci-flat if $$\inf_g||\textrm{Ric}_g||_\infty\cdot \textrm{diam}_g(M)^2=0$$ where $\textrm{Ric}_g$ and $\textrm{diam}_g$ denote the Ricci tensor and the diameter of $g$ respectively and $g$ runs over all Riemannian metrics on $M$. By using Kummer-type method we construct a smooth closed almost Ricci-flat nonspin 5-manifold $M$ which is simply connected. It's minimal volume vanishes, namely it collapses with sectional curvature bounded.

math.DG↗

arXiv2021

The Weyl functional on 4-manifolds of positive Yamabe invariant

It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$\int_M|W^+_g|^2dμ_{g}\geq 2π^2(2χ(M)+3τ(M))-\frac{8π^2}{|π_1(M)|},$$ where $W^+_g$, $χ(M)$ and $τ(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gursky's inequality \cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of $b_2^+(M)>0$ or $δ_gW^+_g=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.

math.DG↗

arXiv2014

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove that there exists a 6-dimensional smooth manifold $M$ with more than one deformation equivalence classes with different signs of $Z(M, [[ω]] )$. Using $Z$ invariants, we set up a Kazdan-Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold $(M, ω)$ of dimension $\geq 4$, any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to $ω$.

math.DG↗

arXiv2014

Finite group actions and G-monopole classes on smooth 4-manifolds

On a smooth closed oriented $4$-manifold $M$ with a smooth action by a compact Lie group $G$, we define a $G$-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$ structure which has a solution of the Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We find $\Bbb Z_k$-monopole classes on some $\Bbb Z_k$-manifolds such as the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant or Bauer-Furuta invariant, where the $\Bbb Z_k$-action is a cyclic permutation of $k$ summands. As an application, we produce infinitely many exotic non-free actions of $\Bbb Z_k\oplus H$ on some connected sums of finite number of $S^2\times S^2$, $\Bbb CP_2$, $\overline{\Bbb CP}_2$, and $K3$ surfaces, where $k\geq 2$, and $H$ is any nontrivial finite group acting freely on $S^3$.

math.GT↗

arXiv2014

G-monopole invariants on some connected sums of 4-manifolds

On a smooth closed oriented $4$-manifold $M$ with a smooth action of a finite group $G$ on a Spin$^c$ structure, $G$-monopole invariant is defined by "counting" $G$-invariant solutions of Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We compute $G$-monopole invariants on some $G$-manifolds. For example, the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant has nonzero $\Bbb Z_k$-monopole invariant mod 2, where the $\Bbb Z_k$-action is given by cyclic permutations of $k$ summands.

math.GT↗

arXiv2014

Some refined higher type adjunction inequalities on 4-manifolds

We further sharpen higher type adjunction inequalities of P. Ozsváth and Z. Szabó on a 4-manifold $M$ with a nonzero Seiberg-Witten invariant for a Spin$^c$ structure $\frak{s}$, when an embedded surface $Σ\subset M$ satisfies $[Σ]\cdot [Σ]\geq 0$ and $$|\langle [Σ],c_1(\frak{s})\rangle|+[Σ]\cdot [Σ]\geq 2b_1(M).$$

math.GT↗

arXiv2013

The symmetry of Spin^c Dirac spectrums on Riemannian product manifolds

It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq 1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or $(e^{\frac{1}{2}c_{1}(L_{1})} \hat{A}(M_1))[M_{1}]=0$, where $L_1$ is the associated line bundle for the given $\text{spin}^{\mathbb{C}}$ structure of $M_1$.

math.DG↗

arXiv2013

An Omori-Yau maximum principle for semi-elliptic operators and Liouville-type theorems

We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold $M$ to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued $C^{2}$ function $f$ on $M$ satisfying $L f \geq F(f)+ H(|\nabla f|) $ for real-valued continuous functions $F$ and $H$ on $\Bbb R$ such that $H(0)=0$.

math.DG↗

arXiv2013

G-monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds

On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological obstruction to the existence of a $G$-invariant nonsingular solution to the normalized Ricci flow on $M$. In particular, for certain $m$ and $n$, $m\Bbb CP_2 # n\bar{\Bbb CP}_2$ admits an infinite family of topologically equivalent but smoothly distinct non-free actions of $\Bbb Z_d$ such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where $d>1$ is a non-prime integer. We also compute the $G$-Yamabe invariants of some 4-manifolds with $G$-monopole classes and the oribifold Yamabe invariants of some 4-orbifolds.

math.DG↗

arXiv2012

Ricci curvature and monopole classes on 3-manifolds

We prove an L^2-estimate involving Ricci curvature and a harmonic 1-form on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on some special connected sums.

math.DG↗

arXiv2011

Connected sums with HP^n or CaP^2 and the Yamabe invariant

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp l \Bbb HP^k\sharp m \bar{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\sharp l CaP^2\sharp m \bar{CaP^2})=Y(M),$$ where $CaP^2$ is the Cayley plane.

math.DG↗

arXiv2010

T-structure and the Yamabe invariant

The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \hat{A}-genus has zero Yamabe invariant. We generalize it to various T-structured manifolds, for example T^m-bundles over such B whose transition functions take values in Sp(m,Z) (or Sp(m-1,Z)\oplus \pm 1 for odd m).

math.DG↗

arXiv2010

On the nonexistence of Einstein metric on 4-manifolds

By using the gluing formulae of the Seiberg-Witten invariant, we show the nonexistence of Einstein metric on manifolds obtained from a 4-manifold with nontrivial Seiberg-Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed oriented 4-manifolds with negative definite intersection form.

math.DG↗