AI 中文总结
该论文推进张量\(\mathrm{H}\)-结构流理论，比较两种演化，证明多种情况下流的短时存在性和唯一性，处理\(\mathrm{SU}(m)\)等情况，得到修正里奇 - 调和流的相关方程和估计，在六维转化为\(\mathrm{SU}(3)\)-结构标准形式描述流族。
AI 中文摘要
我们推进了张量\(\mathrm{H}\)-结构流的一般理论，重点关注非等距流以及\(\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)\)的情况。在发展了相关的\(\mathrm{SU}(m)\)代数后，比较了两种自然演化：内在挠率能量的无限制负梯度流和里奇 - 调和流。证明了对于\(\mathrm{SO}(n)\)的每个闭子群\(\mathrm{H}\)，具有任意低阶挠率二次项的里奇 - 调和\(\mathrm{H}\)-流的短时存在性和唯一性。对于某些特定群，将负梯度流表示为里奇 - 调和形式并证明其短时存在性和唯一性。通过单独的主符号计算处理\(\mathrm{SU}(m)\)情况，还得到了修正里奇 - 调和流的相关演化方程和估计等。在六维中，将形式转化为\(\mathrm{SU}(3)\)-结构的标准挠率形式并描述了相应的二阶拟线性\(\mathrm{SU}(3)\)-流族。
英文摘要
We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.
Comments58 pages, 1 figure. Comments are welcome
