AI 中文总结
研究黎曼曲面上可求长曲线的弱弹性能量，通过内接测地多边形的p-旋转概念经松弛定义该能量，能检测曲线弧长参数化的内在二阶Sobolev正则性，且有限时与测地曲率p次幂积分一致。
AI 中文摘要
我们为紧致无边界可定向光滑黎曼曲面上的可求长曲线引入了一种弱弹性能量。该能量通过从内接测地多边形的p-旋转概念出发进行松弛来定义。此概念通过归一化等温坐标下的局部构造获得。对于每个p>1的指数，所得的松弛泛函精确检测曲线弧长参数化的内在二阶Sobolev正则性。此外，当松弛能量有限时，它与测地曲率的p次幂积分一致。
英文摘要
We introduce a weak elastic energy for rectifiable curves on compact orientable smooth Riemannian surfaces without boundary. The energy is defined by relaxation starting from a notion of $p$-rotation of inscribed geodesic polygonals, that is obtained by a local construction in normalized isothermal coordinates. For every exponent $p>1$, the resulting relaxed functional detects precisely the intrinsic second-order Sobolev regularity of the arc-length parameterization of the curve. Furthermore, when the relaxed energy is finite, it agrees with the integral of the $p$-power of the geodesic curvature.
Comments18 pages, 1 figure
