AI 中文总结
研究吉尔 - 梅德拉诺关于实射影子空间在伯格形变下极小性及体积最小化问题，通过斜伴随自同态对线性几何的控制给出分类，确定体积泛函临界点等，还计算同调短程线并提出相关猜想。
AI 中文摘要
吉尔 - 梅德拉诺提出问题：在大于三维的情况下，实射影子空间在伯格形变下是否仍保持极小性和体积最小化？我们回答了极小性问题，并在相应的最小化问题上取得了重大进展。基本观察是实子空间\(V\subset\mathbb{C}^N\)的全线性几何由单个斜伴随自同态\(A_V = \operatorname{pr}_VJ|_V\)控制。对于每个非圆伯格度量，给出了完整分类：\(\mathbb{RP}(V)\)是极小的当且仅当\(V\)是线性CR子空间，等价于其所有多重凯勒角为\(0\)或\(\pi/2\)。所以吉尔 - 梅德拉诺极小性问题的答案一般是否定的，不过每个赤道超曲面仍保持极小性。相同结构确定了线性体积泛函的临界点和全局极值。结合射影积分几何，我们还计算了纯拉伸和挤压状态下的精确模二同调短程线。最后我们猜想其余情况由混合CR赤道组成。
英文摘要
Gil-Medrano asked whether real projective subspaces remain minimal and volume-minimising under Berger deformations in dimensions greater than three. We answer the minimality question and make substantial progress on the corresponding minimisation problem. The fundamental observation is that the full linear geometry of a real subspace $V\subset\mathbb{C}^N$ is controlled by the single skew-adjoint endomorphism $A_V=\operatorname{pr}_VJ|_V$. For every non-round Berger metric, this yields a complete classification: $\mathbb{RP}(V)$ is minimal if and only if $V$ is a linear CR subspace, equivalently if all of its multiple Kähler angles are $0$ or $π/2$. Thus the answer to Gil-Medrano's minimality question is negative in general, although every equatorial hypersurface remains minimal. The same structure determines the critical points and global extrema of the linear volume functional. Combining it with projective integral geometry, we also compute exact mod-two homological systoles in the pure stretched and squashed regimes. Our final contribution is a conjecture that all remaining cases consist of mixed CR equators.
Comments16 pages
