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Fusheng Deng

Publications and source records attributed to Fusheng Deng.

At least 19 recordsLinked to original sources
arXiv2026

Holomorphic Approximation for Real Diffeomorphism Groups

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

math.CV↗
arXiv2025

Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact K\"ahler manifolds

The main goal of this paper is to generalize the Sobolev-type inequalities given by Guo-Phong-Song-Sturm and Guedj-T\^o from the case of functions to the framework of twisted differential forms. To this end, we establish certain estimates of heat kernels for differential forms with values in holomorphic vector bundles over compact K\"ahler manifolds. As applications of these estimates, we also prove a vanishing theorem and give certain $L^{q,p}$-estimates for the $\bar\partial$-operator on twisted differential forms.

math.CV↗
arXiv2025

Log truncated threshold and zero mass conjecture

For plurisubharmonic functions $\varphi$ and $\psi$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{\varphi(z), \psi(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $\varphi$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $\varphi$ at $0$ which reflects a singular property of $\varphi$ via a log function near the origin (denoted by $lt(\varphi,0)$) and derive an optimal estimate of the residual Monge-Amp\`ere mass of $\varphi$ at $0$ in terms of its higher order Lelong numbers $\nu_j(\varphi)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(\varphi,0)<\infty$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture.

math.CV↗
arXiv2025

Dirichlet Green kernel estimates and Sobolev-type inequalities for twisted differential forms

We study the full-trace Dirichlet realization of the Dolbeault Laplacian on differential forms with values in a Hermitian holomorphic vector bundle over a relatively compact smooth domain in a K\"ahler manifold. We prove global Green kernel estimates that are uniform up to the boundary, including one- and two-boundary-factor bounds and estimates for the \(\bar\partial\)- and \(\bar\partial^*\)-derivatives. These estimates yield Sobolev-type inequalities with boundary terms. In real dimension two, the first-order bound has a logarithmic loss. In top antiholomorphic degree, we obtain quantitative \(L^r\)-to-\(L^k\) solvability for \(\bar\partial\) without pseudoconvexity. Analogous results hold for the twisted de Rham complex of a flat metric connection on a compact Riemannian manifold with smooth boundary.

math.AP↗
arXiv2024

Uniform estimates of Green functions and Sobolev-type inequalities on real and complex manifolds

We prove certain $L^p$ Sobolev-type and Poincar\'e-type inequalities for functions on real and complex manifolds for the gradient operator $\nabla$, the Laplace operator $\Delta$, and the operator $\bar\partial$. Integral representations for functions are key to get such inequalities. The proofs of the main results involves certain uniform estimates for the Green functions and their gradients on Riemannian manifolds, which are also established in the present work.

math.CV↗
arXiv2024

Uniqueness of irreducible desingularization of singularities associated to negative vector bundles

We prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two negative line bundles over compact complex manifolds are isomorphic if and only if their Grauert blow downs have isomorphic germs near the singularities. We also show that there is a unique way to modify a submanifold of a complex manifold to a hypersurface, namely, the blow up of the ambient manifold along the submanifold.

math.AG↗
arXiv2024

Multiplicites and modifications, and singularities associated to blowing down negative vector bundles

We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.

math.CV↗
arXiv2024

$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and apply them to derive a generalization of the Sobolev Inequality with Trace in $\mathbb{R}^n$. As applications to complex analysis, we get an integral form of Maximum Modulus Principle for holomorphic functions, and an improvement of H\"ormander's $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains.

math.CV↗
arXiv2023

Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.

math.CV↗
arXiv2022

Curvature strict positivity of direct image bundles associated to pseudoconvex families of domains

We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains. The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular domains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic. This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis. We investigate that the main result implies a remarkable result of Berndtsson which states that, for an ample vector bundle $E$ over a compact complex manifold $X$ and any $k\geq 0$, the bundle $S^kE\otimes\det E$ admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where $S^kE$ is the $k$-th symmetric product of $E$. The two main ingredients in the argument of the main theorems are Berndtsson's estimate of the lower bound of curvature of direct image bundles and Deng-Ning-Wang-Zhou's characterization of the curvature Nakano positivity of Hermitian vector bundles in terms of $L^2$-estimate of $\bar\partial$.

math.CV↗
arXiv2022

Linear isometric invariants of bounded domains

We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.

math.CV↗
arXiv2021

Omnidirectional nonreciprocal absorber realized by the magneto-optical hypercrystal

Photonic bandgap design is one of the most basic ways to effectively control the interaction between light and matter. However, the traditional photonic bandgap is always dispersive (blueshift with the increase of the incident angle), which is disadvantageous to the construction of wide-angle optical devices. Hypercrystal, that the photonic crystal with layered hyperbolic metamaterials (HMMs), can strongly modify the bandgap properties based on the anomalous wavevector dispersion of the HMM. Here, based on phase variation compensation between HMM and isotropic dielectric layers, we propose for the first time to design nonreciprocal and flexible photonic bandgaps using magneto-optical HMMs in one-dimensional photonic crystals. Especially for the forward and backward incident light, the blueshift and dispersionless of the forward and backward cavity modes are designed respectively to realize the interesting omnidirectional nonreciprocal absorber. Our results show high (low) absorption about 0.99 (0.25) in an angle range of 20-75 degrees for the forward (backward) incident light at the wavelength of 367 nm. The nonreciprocal omnidirectional cavity mode not only facilitates the design of perfect unidirectional optical absorbers working in a wide-angle range, but also possesses significant applications for all-angle reflectors and filters.

physics.optics↗
arXiv2020

Curvature positivity of invariant direct images of Hermitian vector bundles

We prove that the invariant part, with respect to a compact group action satisfying certain condition, of the direct image of a Nakano positive Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is Nakano positive. We also consider the action of the noncompact group $\mathbb{R}^m$ and get the same result for a family of tube domains, which leads to a new method to the matrix-valued Prekopa's theorem originally proved by Raufi. The two main ingredients in our method are Hörmander's $L^2$ theory of $\bar\partial$ and the recent work of Deng-Ning-Zhang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles.

math.CV↗
arXiv2020

Characterization of Curvature positivity of Riemannian metrics on flat vector bundles

We give a characterization of Nakano positivity of Riemannian flat vector bundles over bounded domains $D\subset\mathbb{R}^n$ in terms of solvability of the $d$ equation with certain good $L^2$ estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.

math.CV↗
arXiv2020

Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$

We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension condition for some $p>0$. These results can be roughly viewed as converses of Hörmander's $L^2$-estimate of $\bar\partial$ and Ohsawa-Takegoshi type extension theorems. As an application of the main result, we get a totally different method to Nakano positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic families of complex manifolds.

math.CV↗