关于构造双调和与共形双调和映射到球面的remarks
Remarks on constructing biharmonic and conformal biharmonic maps to spheres
AI总结：
本文提出一种几何算法，用于构造双调和与共形双调和映射到球面，并通过实例展示算法在不同域下的灵活性及方法的有效性。
摘要：
双调和与共形双调和映射是Riemann几何中已研究广泛的调和映射的第四次推广。在本文中，我们考虑映射到欧几里得球面的情况，并研究一种几何算法，旨在使给定的调和映射成为双调和或共形双调和映射。对于双调和映射，我们发现当域为闭合时，最大原理对我们的方法施加了严格的限制，而在非紧域的情况下则有更多灵活性，并通过多个例子突显这种差异。关于共形双调和映射，我们展示了我们的算法可以产生球面之间的映射的显式临界点。此外，我们发现与双调和映射不同，我们没有得到严格的限制，因此我们的算法可能产生额外的球面之间的共形双调和映射，而不仅仅是本文中找到的那些。
英文摘要：
Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric algorithm that aims at rendering a given harmonic map either biharmonic or conformally biharmonic. For biharmonic maps we find that in the case of a closed domain the maximum principle imposes strong restrictions on our approach, whereas there is more flexibility when we have a non-compact domain and we highlight this difference by a number of examples. Concerning conformal-biharmonic maps we show that our algorithm produces explicit critical points for maps between spheres. Moreover, it turns out that we do not get strong restrictions as we obtain for biharmonic maps, such that our algorithm might produce additional conformal-biharmonic maps between spheres beyond the ones found in this article.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
