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Jih-Hsin Cheng

Publications and source records attributed to Jih-Hsin Cheng.

At least 19 recordsLinked to original sources
arXiv2026

Positive mass theorem and the Yamabe equation on CR manifolds

Our goal is to survey the development of positive mass theorem and the Yamabe equation on CR manifolds in recent years. We introduce the notion of the mass in several complex variables or CR geometry. We then consider the Yamabe problem on CR manifolds to find a minimizer for the CR-Sobolev quotient. The positive mass theorem plays a key role in finding a solution to the Yamabe equation with minimum energy for the positive curvature case. We mainly focus on the team works in the following three papers [CMY17], [CMY23] and [CC22], on a positive mass theorem in 3-dimensional CR geometry, the CR-Sobolev quotient of Rossi spheres, and the 5-dimensional situation, respectively.

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arXiv2025

On minimizing surfaces of the CR invariant energy $E_1$

We study a CR-invariant equation for vanishing $E_1$ surfaces in the 3-dimensional Heisenberg group. This is shown to be a hyperbolic equation. We prove the local uniqueness theorem for an initial value problem and classify all such global surfaces with rotational symmetry. We also show that the Clifford torus in the CR 3-sphere is not a local minimizer of $E_1$ by computing the second variation.

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arXiv2025

Spectral bundles on Abelian varieties, complex projective spaces and Grassmannians

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.

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arXiv2024

Heat kernel and local index theorem for open complex manifolds with $\mathbb{C}^{\ast }$-action

For a complex manifold $\Sigma $ with $\mathbb{C}^{\ast }$-action, we define the $m$-th $\mathbb{C}^{\ast }$ Fourier-Dolbeault cohomology group and consider the $m$-index on $\Sigma $. By applying the method of transversal heat kernel asymptotics, we obtain a local index formula for the $m$-index. We can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a compact complex orbifold with an orbifold holomorphic line bundle by our integral formulas over a (smooth) complex manifold and finitely many complex submanifolds arising from singular strata. We generalize $\mathbb{C}^{\ast }$-action to complex reductive Lie group $G$-action on a compact or noncompact complex manifold. Among others, we study the nonextendability of open group action and the space of all $G$-invariant holomorphic $p$-forms. Finally, in the case of two compatible holomorphic $\mathbb{C}^{\ast }$-actions, a mirror-type isomorphism is found between two linear spaces of holomorphic forms, and the Euler characteristic associated with these spaces can be computed by our $\mathbb{C}^{\ast }$ local index formula on the total space. In the perspective of the equivariant algebraic cobordism theory $\Omega _{\ast }^{\mathbb{C}^{\ast }}(\Sigma ),$ a speculative connection is remarked. Possible relevance to the recent development in physics and number theory is briefly mentioned.

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arXiv2023

On the variation of the Einstein-Hilbert action in pseudohermitian geometry

In this paper we compute the first and second variation of the normalized Einstein-Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater or equal to five, in three dimension we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

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arXiv2022

Theta functions and adiabatic curvature on an Abelian variety

For an ample line bundle $L$ on an Abelian variety $M$, we study the theta functions associated with the family of line bundles $L\otimes T$ on $M$ indexed by $T\in \text{Pic}^{0}(M)$. Combined with an appropriate differential geometric setting, this leads to an explicit curvature computation of the direct image bundle $E$ on $\text{Pic}^{0}(M)$, whose fiber $E_{T}$ is the vector space spanned by the theta functions for the line bundle $L\otimes T$ on $M$. Some algebro-geometric properties of $E$ are also remarked.

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arXiv2021

Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except dimension 5 (n = 2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.

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arXiv2019

Connected sum of CR manifolds with positive CR Yamabe constant

Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.

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arXiv2019

Chains in CR geometry as geodesics of a Kropina metric

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspondence between geodesics of the Kropina metric and chains allows us to use the methods of metric geometry and the calculus of variations to study chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985] that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain. Finally, we generalize this result to the global setting by showing that any two points of a connected compact strictly pseudoconvex CR manifold which admits a pseudo-Einstein contact form with positive Tanaka-Webster scalar curvature can be joined by a chain.

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arXiv2019

Theta Functions and Adiabatic Curvature on a Torus

Let $M$ be a complex torus, $L_{\hatμ}\to M$ be positive line bundles parametrized by $\hat μ\in {\rm Pic}^0(M)$, and $E\to {\rm Pic}^0(M)$ be a vector bundle with $E|_{\hatμ}\cong H^0(M, L_{\hat μ})$. We endow the total family $\{L_{\hatμ}\}_{\hatμ}$ with a Hermitian metric that induces the $L^2$-metric on $H^0(M, L_{\hat μ})$ hence on $E$. By using theta functions $\{θ_m\}_{m}$ on $M\times M$ as a family of functions on the first factor $M$ with parameters in the second factor $M$, our computation of the full curvature tensor $Θ_E$ of $E$ with respect to this $L^2$-metric shows that $Θ_E$ is essentially an identity matrix multiplied by a constant $2$-form, which yields in particular the adiabatic curvature $c_1(E)$. After a natural base change $M\to \hat M$ so that $E\times_{\hat M} M:=E'$, we also obtain that $E'$ splits holomorphically into a direct sum of line bundles each of which is isomorphic to $L_{\hatμ=0}^*$. Physically, the spaces $H^0(M, L_{\hat μ})$ correspond to the lowest eigenvalue with respect to certain family of Hamiltonian operators on $M$ parametrized by $\hatμ$ or in physical notation, by wave vectors $\bf k$.

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arXiv2019

On the Sobolev quotient of three-dimensional CR manifolds

We exhibit examples of compact three-dimensional CR manifolds of positive Webster class, {\em Rossi spheres}, for which the pseudo-hermitian mass as defined in \cite{CMY17} is negative, and for which the infimum of the CR-Sobolev quotient is not attained. To our knowledge, this is the first geometric context on smooth closed manifolds where this phenomenon arises, in striking contrast to the Riemannian case.

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arXiv2017

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.

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arXiv2017

Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action

Among those transversally elliptic operators initiated by Atiyah and Singer, Kohn's $\Box_b$ operator on CR manifolds with $S^1$ action is a natural one of geometric significance for complex analysts. Our first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves an unprecedented contribution in terms of a distance function from lower dimensional strata of the $S^1$-action. Our second main result computes a local index density, in terms of \emph{tangential} characteristic forms, on such manifolds including \emph{Sasakian manifolds} of interest in String Theory, by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler's rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds. As applications of our CR index theorem we can prove a CR version of Grauert-Riemenschneider criterion, and produce many CR functions on a weakly pseudoconvex CR manifold with transversal $S^1$ action and many CR sections on some class of CR manifolds, answering (on this class of manifolds) some long-standing questions in several complex variables and CR geometry. We give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover in some cases, without use of equivariant cohomology method nor keeping contributions arising from lower dimensional strata as done in previous works, we can reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold with an orbifold holomorphic line bundle, as an index theorem obtained by a single integral over a smooth CR manifold which is essentially the circle bundle of this line bundle.

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arXiv2016

Strong maximum principle for mean curvature operators on subriemannian manifolds

We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher dimensions for two cases: (a) the touching point is nonsingular; (b) the touching point is an isolated singular point for one of comparison functions. For a background subriemannian manifold with local symmetry of isometric translations, we have the strong maximum principle for associated graphs which include, among others, intrinsic graphs with constant horizontal (p-) mean curvature. As applications, we show a rigidity result of horizontal (p-) minimal hypersurfaces in any higher dimensional Heisenberg cylinder and a pseudo-halfspace theorem for any Heisenberg group.

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arXiv2015

Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.

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