TheauthorobtainsaWeierstrassrepresentationforsurfaceswithprescribednormalGaussmapandGausscurvatureinH3.AdifferentialequationaboutthehyperbolicGaussmapisalsoobtained,whichcharacterizestherelationamongthehyperbolicGaussmap,thenormalGaussmapandGausscurvature.TheauthordiscussestheharmonicityofthenormalGaussmapandthehyperbolicGaussmapfromsurfacewithconstantGausscurvatureinH3toS2withcertainalteredconformalmetric.Finally,theauthorconsidersthesurfacewhosenormalGaussmapisconformalandderivesacompletelynonlineardifferentialequationofsecondorderwhichgraphmustsatisfy.