Searcharxiv⌕ Search

arXiv subjects

Shihshu Walter Wei

Publications and source records attributed to Shihshu Walter Wei.

16 recordsLinked to original sources
arXiv2023

Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications

In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to $p$-subharmonicity, subsolutions to the $p$-Laplace equation, uniqueness, existence, isometric immersions in multiple settings. In particular, we show that a convex function on $\mathbb{R}^n$ is a $p$-subharmonic function, for every $p > 1$, and a $C^2$ convex function on a Riemannian manifold is a $p$-subharmonic function $f$, for every $p > 1\, .$ We also show that a $C^2$ convex function which is a submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p \ge 1\, .$ This result is sharp. As further applications, via function growth estimates in $p$-harmonic geometry, we prove that every $p$-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for $p > 1$. In particular, every $L^q$, nonnegative, convex function of class $C^2$ on a complete noncompact Riemannian manifold is constant for $q > p -1 > 0\, .$

math.AP↗
arXiv2023

$n$-Harmonicity, Minimality, Conformality and Cohomology

By studying cohomology classes that are related with $n$-harmonic morphisms and $F$-harmonic maps, we augment and extend several results on $F$-harmonic maps, harmonic maps in [1, 3, 14], $p$-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and $n$-harmonic morphisms which are submersions. The results, for example Theorem 3.2 obtained by utilizing the $n$-conservation law (2.6), are sharp.

math.DG↗
arXiv2023

The geometry of $Φ_{(3)}$-harmonic maps

In this paper, we motivate and extend the study of harmonic maps or $Φ_{(1)}$-harmonic maps (cf [15], Remark 1.3 (iii)), $Φ$-harmonic maps or $Φ_{(2)}$-harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of $Φ_{(3)}$-harmonic maps by unified geometric analytic methods. We define the notion of $Φ_{(3)}$-harmonic maps and obtain the first variation formula and the second variation formula of the $Φ_{(3)}$-energy functional $E_{Φ_{(3)}}$. By using a stress-energy tensor, the $Φ_{(3)}$-conservation law, a monotonicity formula, and the asymptotic assumption of maps at infinity, we prove Liouville type results for $Φ_{(3)}$-harmonic maps. We introduce the notion of $Φ_{(3)}$-Superstrongly Unstable ($Φ_{(3)}$-SSU) manifold and provide many interesting examples. By using an extrinsic average variational method in the calculus of variations (cf. [51, 49]), we find $Φ_{(3)}$-SSU manifold and prove that for $i=1,2,3$, every compact $Φ_{(i)}$-$\operatorname{SSU}$ manifold is $Φ_{(i)}$-$\operatorname{SU}$, and hence is $Φ_{(i)}$-$\operatorname{U}$ (cf. Theorem 9.3). As consequences, we obtain topological vanishing theorems and sphere theorems by employing a $Φ_{(3)}$-harmoic map as a catalyst. This is in contrast to the approaches of utilizing a geodesic ([45]), minimal surface, stable rectifiable current ([34, 29, 50]), $p$-harmonic map (cf. [53]), etc., as catalysts. These mysterious phenomena are analogs of harmonic maps or $Φ_{(1)}$-harmonic maps, $p$-harmonic maps, $Φ_{S}$-harmonic maps, $Φ_{S,p}$-harmonic maps, $Φ_{(2)}$-harmonic maps, etc., (cf. [21, 40, 42, 41, 12, 13]).

math.DG↗
arXiv2023

Some links between $F$-harmonicity, submersion and cohomology

By studying cohomology classes that are related with $p$-harmonic morphisms, $F$-harmonic maps, and $f$-harmonic maps, we extend several of our previous results on Riemannian submersions and $p$-harmonic morphisms to $F$-harmonic maps, and $f$-harmonic maps which are submersions.

math.DG↗
arXiv2022

On exponential Yang-Mills fields and $p$-Yang-Mills fields

We introduce \emph{normalized exponential Yang-Mills energy functional} $\mathcal{YM}_e^0$, stress-energy tensor $S_{e,\mathcal{YM}^0 }$ associated with the normalized \emph{exponential Yang-Mills energy functional} $\mathcal{YM}_e ^0 $, $e$-conservation law. We also introduce the notion of the {\it $e$-degree} $d_e$ which connects two separate parts in the associated normalize exponential stress-energy tensor $S_{e,\mathcal{YM}^0 }$ (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang-Mills fields, and prove a vanishing theorem for exponential Yang-Mills fields. These monotonicity formula and vanishing theorem for exponential Yang-Mills fields augment and extend monotonicity formula and vanishing theorem for $F$-Yang-Mills fields in [DW] and [W11, 9.2]. We also discuss an average principle (cf. Proposition 8.1), isoperimetric and Sobolev inequalities, convexity and Jensen's inequality, $p$-Yang-Mills fields, an extrinsic average variational method in the calculus of variation (cf.[W1, W3]) and $Φ_{(3)}$-harmonic maps, from varied, coupled, generalized viewpoints and perspectives (cf. Theorems 6.1, 7.1, 9.1, 9.2, 10.1,10.2, 11.13, 11.14, 11.15)).

math.DG↗
arXiv2021

Dualities in Comparison Theorems and Bundle-Valued Generalized Harmonic Forms on Noncompact Manifolds

We observe, utilize dualities in differential equations and differential inequalities, dualities between comparison theorems in differential equations, and obtain dualities in "swapping" comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II and lead to comparison theorems in Riemannian geometry with analytic, geometric, P.D.E.'s and physical applications. In particular, we prove Hessian comparison theorems and Laplacian comparison theorem under varied radial Ricci curvature or radial curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei, and Wei. We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship. These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving duality theorem and unity theorem, generalizing the work of Andreotti and Vesentini, and Wei. We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities, embedding theorem for weighted Sobolev spaces, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree $k$ with values in vector bundles, such as $F$-Yang Mills fields (when $F$ is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for $F$-harmonic maps, and Dirichlet problems on starlike domains for vector bundle valued differential $1$-forms and $F$-harminic maps, etc.

math.DG↗
arXiv2020

Growth Estimates for Generalized Harmonic Forms on Noncompact Manifolds with Geometric Applications

We introduce Condition W $\,$(1.2) for a smooth differential form $ω$ on a complete noncompact Riemannian manifold $M$. We prove that $ω$ is a harmonic form on $M$ if and only if $ω$ is both closed and co-closed on $M\, ,$ where $ω$ has $2$-balanced growth either for $q=2$, or for $1 < q(\ne 2) < 3\, $ with $ω$ satisfying Condition W $\,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1<q(\ne 2)<3\, $ satisfying Condition W $\,$(1.2) is both closed and co-closed (cf. Theorem 1.1). This generalizes the work of A. Andreotti and E. Vesentini [AV] for every $L^2$ harmonic form $ω\, .$ In extending $ω$ in $L^2$ to $L^q$, for $q \ne 2$, Condition W $\,$(1.2) has to be imposed due to counter-examples of D. Alexandru-Rugina$\big($ [AR] p. 81, Remarque 3$\big).$ We then study nonlinear partial differential inequalities for differential forms $ \langleω, Δω\rangle \ge 0, $ in which solutions $ω$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $ω\, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $\langleω, Δω\rangle \ge 0\, ,$ (ii) $Δω= 0\, ,$ $($iii$)$$\quad d\, ω= d^{\star}ω= 0\, ,$ (iv) $\langle \star\, ω, Δ\star\, ω\rangle \ge 0\, ,$ (v) $Δ\star\, ω= 0\, ,$ and (vi) $d\, \star\, ω= d^{\star} \star\, ω= 0\, $ are equivalent (cf. Theorem 4.1). We also study As geometric applications, we employ the theory in [DW] and [W3], solve constant Dirichlet problems for generalized harmonic $1$-forms and $F$-harmomic maps (cf. Theorems 10.3 and 10.2), derive monotonicity formulas for $2$-balanced solutions, and vanishing theorems for $2$-moderate solutions of $\langleω, Δω\rangle \ge 0\, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).

math.DG↗
arXiv2019

$Φ$-Harmonic Maps and $Φ$-Superstrongly Unstable Manifolds

In this paper, we motivate and define $Φ$-energy density, $Φ$-energy, $Φ$-harmonic maps and stable $Φ$-harmonic maps. Whereas harmonic maps or $p$-harmonic maps can be viewed as critical points of the integral of $σ_1$ of a pull-back tensor, $Φ$-harmonic maps can be viewed as critical points of the integral of $σ_2$ of a pull-back tensor. By an extrinsic average variational method in the calculus of variations (cf. \cite{HW,WY,13,HaW}), we derive the average second variation formulas for $Φ$-energy functional, express them in orthogonal notation in terms of the differential matrix, and find $Φ$-superstrongly unstable $(Φ$-$\text{SSU})$ manifolds. We prove, in particular that every compact $Φ$-$\text{SSU}$ manifold must be $Φ$-strongly unstable $(Φ$-$\text{SU})$, i.e., $(a)$ A compact $Φ$-$\text{SSU}$ manifold cannot be the target of any nonconstant stable $Φ$-harmonic maps from any manifold, $\rm (b)$ The homotopic class of any map from any manifold into a compact $Φ$-$\text{SSU}$ manifold contains elements of arbitrarily small $Φ$-energy, $(\rm c)$ A compact $Φ$-$\text{SSU}$ manifold cannot be the domain of any nonconstant stable $Φ$-harmonic map into any manifold, and $(\rm d)$ The homotopic class of any map from a compact $Φ$-$\text{SSU}$ manifold into any manifold contains elements of arbitrarily small $Φ$-energy (cf. Theorem $1.1 (a),(b),(c)$, and $(d)$.) We also provide many examples of $Φ$-SSU manifolds, and establish a link of $Φ$-SSU manifold to $p$-SSU manifold and topology. The extrinsic average variational method in the calculus of variations that we have employed is in contrast to an average method in PDE that we applied in \cite {CW} to obtain sharp growth estimates for warping functions in multiply warped product manifolds.

math.DG↗
arXiv2018

Sharp growth estimates for warping functions in multiply warped product manifolds

By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.

math.DG↗
arXiv2016

$L^2$ curvature pinching theorems and vanishing theorems on complete Riemannian manifolds

In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar curvature, under curvature pinching conditions. Secondly, we prove vanishing results for $L^2$ and some non-$L^2$ harmonic $p$-forms on LCF manifolds, by assuming that the underlying manifolds satisfy pointwise or integral curvature conditions. Moreover, by a Theorem of Li-Tam for harmonic functions, we show that the underlying manifold must have only one end. Finally, we obtain Liouville theorems for $p$-harmonic functions on LCF manifolds under pointwise Ricci curvature conditions.

math.DG↗
arXiv2016

Liouville properties for p-harmonic maps with finite q-energy

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization $L_ε$ of a perturbed p-Laplace operator. By deriving an $L_ε$-type Bochner's formula and a Kato type inequality, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds, where the range for q contains p. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds.

math.DG↗
arXiv2013

Submanifolds of warped product manifolds $I\times_f S^{m-1}(k)$}} from a $p$-harmonic viewpoint

We study $p$-harmonic maps, $p$-harmonic morphisms, biharmonic maps, and quasiregular mappings into submanifolds of warped product Riemannian manifolds ${I}\times_f S^{m-1}(k)\, $ of an open interval and a complete simply-connecteded $(m-1)$-dimensional Riemannian manifold of constant sectional curvature $k$. We establish an existence theorem for $p$-harmonic maps and give a classification of complete stable minimal surfaces in certain three dimensional warped product Riemannian manifolds ${\bf R}\times_f S^{2}(k)\, ,$ building on our previous work. When $f \equiv\, $ Const. and $k=0$, we recapture a generalized Bernstein Theorem and hence the Classical Bernstein Theorem in $R^3$. We then extend the classification to parabolic stable minimal hypersurfaces in higher dimensions.

math.AP↗
arXiv2010

Generalized $1$-harmonic Equation and The Inverse Mean Curvature Flow

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equations (1.1) on a domain in a Riemannian $n$-manifold that affects the behavior of weak solutions of (1.1), and establish its link with the geometry of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for $p$-subharmonic functions of constant $p$-tension field, $p \ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.

math.DG↗
arXiv2010

On Vanishing Theorems For Vector Bundle Valued p-Forms And Their Applications

Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ function with $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When $F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,$ and $1-\sqrt{1-2t},$ the $F$-Yang-Mills field becomes an ordinary Yang-Mills field, $p$-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) and derive the first variational formula of the $E_{F,g}-$energy functional (resp. $F$-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors, linked to $F$-conservation laws yield monotonicity formulae. A "macroscopic" version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for $p-$forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for $F-$harmonic maps (e.g. $p$-harmonic maps), and $F-$Yang-Mills fields (e.g. generalized Yang-Mills-Born-Infeld fields on manifolds). We also obtain generalized Chern type results for constant mean curvature type equations for $p-$forms on $\Bbb{R}^m$ and on manifolds $M$ with the global doubling property by a different approach. The case $p=0$ and $M=\mathbb{R}^m$ is due to Chern.

math.DG↗
arXiv2007

On 1-Harmonic Functions

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over $\mathbb{R}$; and every 7-dimensional $SO(2)\times SO(6)$-invariant absolutely area-minimizing integral current in $\mathbb{R}^8$ is real analytic. The assumption on the $SO(2) \times SO(6)$-invariance cannot be removed, due to the first counter-example in $\mathbb{R}^8$, proved by Bombieri, De Girogi and Giusti.

math.DG↗