We study the moduli space of congruence classes of isometric surfaces withthe same mean curvature in 4-dimensional space forms. Having the same meancurvature means that there exists a parallel vector bundle isometry between thenormal bundles that preserves the mean curvature vector fields. We prove thatif both Gauss lifts of a compact surface to the twistor bundle are notvertically harmonic, then there exist at most three nontrivial congruenceclasses. We show that surfaces with a vertically harmonic Gauss lift possess aholomorphic quadratic differential, yielding thus a Hopf-type theorem. We provethat such surfaces allow locally a one-parameter family of isometricdeformations with the same mean curvature. This family is trivial only if thesurface is superconformal. For such compact surfaces with non-parallel meancurvature, we prove that the moduli space is the disjoint union of two sets,each one being either finite, or a circle. In particular, for surfaces in$\R^4$ we prove that the moduli space is a finite set, under a condition on theEuler numbers of the tangent and normal bundles.
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机译：我们研究等距曲面的全等类在4维空间形式中具有相同平均曲率的模空间。具有相同的平均曲率意味着在正常束之间存在平行的矢量束等轴测图，其中保留了平均曲率矢量场。我们证明，如果紧致曲面到扭束的两个高斯提升都不是垂直谐波，那么最多存在三个非平凡的同余类。我们证明了具有垂直谐波高斯升力的表面具有全同二次平方微分，从而产生了Hopf型定理。我们证明了这样的表面局部允许具有相同平均曲率的一参数等轴测变形族。仅当表面超保形时，该族才是琐碎的。对于这种具有非平行平均曲率的紧致曲面，我们证明了模空间是两组不相交的并集，每组要么是有限的，要么是一个圆。特别地，对于在\ R ^ 4 $中的曲面，我们证明在切线束和法线束的欧拉数的条件下，模空间是有限集。
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