Spin(7) Instantons and Hermitian Yang–Mills Connections for the Stenzel Metric

We use the highly symmetric Stenzel Calabi–Yau structure on $$T^{\star }S^{4}$$ T ⋆ S 4 as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This is comprised of two 1-parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a removable singularity phenomenon that we aim to further examine in future work.

Classification of positive radial solutions to a weighted biharmonic equation

<p style='text-indent:20px;'>In this paper, we consider the weighted fourth order equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 5 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ -n&lt;\alpha&lt;n-4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ (p,\alpha,\beta,n) $\end{document}</tex-math></inline-formula> belongs to the critical hyperbola</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of radial solutions to the equation for some <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. On the other hand, let <inline-formula><tex-math id="M7">\begin{document}$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ t = -\ln |x| $\end{document}</tex-math></inline-formula>, then for the radial solution <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula> with non-removable singularity at origin, <inline-formula><tex-math id="M10">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is a periodic function if <inline-formula><tex-math id="M11">\begin{document}$ \alpha \in (-2,n-4) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> satisfy some conditions; while for <inline-formula><tex-math id="M14">\begin{document}$ \alpha \in (-n,-2] $\end{document}</tex-math></inline-formula>, there exists a radial solution with non-removable singularity and the corresponding function <inline-formula><tex-math id="M15">\begin{document}$ v(t) $\end{document}</tex-math></inline-formula> is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.</p>

Structure theorems for singular minimal laminations

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of {\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in {\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

On Removable Singularities of Solutions of Higher-Order Differential Inequalities

We obtain sufficient conditions for solutions of the mth-order differential inequality\sum_{|\alpha|=m}\partial^{\alpha}a_{\alpha}(x,u)\geq f(x)g(|u|)\quad\text{in % }B_{1}\setminus\{0\}to have a removable singularity at zero, where {a_{\alpha}}, f, and g are some functions, and {B_{1}=\{x:|x|<1\}} is a unit ball in {{\mathbb{R}}^{n}}. We show in some examples the sharpness of these conditions.

Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

We study isolated singularities of 2D Yang–Mills–Higgs (YMH) fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general, the singularity cannot be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the YMH field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of 2D harmonic maps.

Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

We study the semilinear elliptic equation -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where {B_{1}\subset{\mathbb{R}}^{n}} , with {n\geq 3} , {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution {u\in C^{2}(B_{1}\setminus\{0\})} of the above equation either has a removable singularity at the origin or it behaves like u(x)=A(1+o(1))|x|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}