We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross section, we show well-posedness of the short-time solution in the \(L^q\)-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blowup of the scalar curvature close to the singularities, we use maximal \(L^q\)-regularity theory for conically degenerate operators.

中文翻译：

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Yamabe流下的圆锥形歧管

我们考虑圆锥奇异流形上的非归一化Yamabe流。在初始横截面的某些几何假设下，我们在\（L ^ q \）设置中显示了短时解的适定性。此外，通过提供在初始局部几何形状附近靠近边界的演化度量的渐近展开，我们给出了在流动条件下圆锥形尖端变形的图片。由于标量曲率的膨胀接近奇点，因此我们对锥简并算子使用最大\（L ^ q \） -正则性理论。