微分几何
The existence and uniqueness of canonical singular solutions of the J-equation and the deformed Hermitian Yang Mills (dHYM) equation was proved in \cite{DMS24} on compact K\"{a}hler surfaces. In this paper, we study the singularity…
We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic…
We construct and give a complete classification of all the differential symmetry breaking operators D_{{\lambda},{\nu}}^m : C^\infty(S^3, V^3_{\lambda}) \rightarrow C^\infty(S^2,L_{m,{\nu}}), between the spaces of smooth sections of a…
A manifold $M$ is said to be a double disk bundle if it can be decomposed as a union of two disk bundles glued together by a diffeomorphism of their boundaries. We show that if $M^n$ is a closed simply connected $n$-manifold with $n$ even…
In this paper we study the non-existence of solutions to the Dirichlet problem for minimal graphs of codimension $\geq 2$, including certain situations over domain $\Omega$ even with non-$C^1$ boundary $\partial \Omega$.
We derive the exact asymptotic rates of the small eigenvalues of the Laplacian on one-parameter degenerations of compact K\"ahler manifolds equipped with induced background metrics. This generalizes a recent result of Dai and Yoshikawa to…
A non-trivial spinor field $\psi$ is called a generalized imaginary $\mathrm{Spin}^c$-Killing spinor if $\nabla^{g,A} _X \psi = i\mu X \cdot \psi$ for all vector fields $X$, where $\mu$ is a real function that is not identically zero and…
We provide a rigorous construction of I.M. Singer's universal connection, a natural connection on a bundle of paths associated to any manifold, using the theory of diffeology. Furthermore, we generalize the universal connection to the…
We study homeomorphisms of the circle that are smooth diffeomorphisms away from a finite set of $n$ points. These "broken diffeomorphisms" do not form a Lie group, but instead naturally assemble into a Lie groupoid. We construct an explicit…
We prove the lossless unit interval Strichartz theorem on asymptotically conic surfaces, assuming that a large enough neighborhood of its trapped set has negative curvature. We also prove the spectral projection theorem on surfaces with…
We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard…
In this work we construct non-holomorphic, complete and minimal submanifolds of the odd-dimensional complex projective spaces $\cn P^{2n-1}$ and their dual complex hyperbolic spaces $\cn H^{2n-1}$. We then provide complete minimal…
This paper investigates the qualitative behavior of a system of ordinary differential equations (ODEs) defined by a matrix operator derived from the algebraic structure of the Alpha Group. The system depends on a rotational parameter that…
A higher extremal K\"ahler metric is defined (motivated by analogy with the definition of an extremal K\"ahler metric) as one whose top Chern form equals a smooth function multiplied by its volume form such that the gradient of the function…
This paper presents a simplified geometric proof of the Molino-Alexandrino-Radeschi (MAR) Theorem, which states that the closure of a singular Riemannian foliation on a complete Riemannian manifold is itself a smooth singular Riemannian…
A timelike splitting theorem for Finsler spacetimes was previously established by the third author, in collaboration with Lu and Minguzzi, under relatively strong hypotheses, including the Berwald condition. This contrasts with the more…
Consider a complete Riemannian manifold $(M, g)$ and optimal transport problems on it with cost functions of the form $c(x,y) = h(d_{{g}}(x,y))$. We study the absolute continuity of the corresponding generalized Wasserstein barycenters of…
We show that intrinsic and extrinsic area density bounds are equivalent, with matching asymptotic values, for complete, connected, smooth minimal immersions $i:\Sigma^d\to\mathbb{R}^N$ of any dimension and codimension. Combining our results…
Given a Lipschitz conductor $K$ in the smooth compact Riemannian $2\le n$-manifold $(M,g)$, such a half generic heat dispersion law $$ {\rm H^d}_{p,\varPhi,\varPsi}(K,M)=2^{-1} {\rm H^d}_{\Delta_p,\varPhi,\varPsi}(K,M) $$ is not only…
Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschl\"ager [{\it Invent. Math.,} 2001], we derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship…