Maybe you have an older version of $\LaTeX$ installed that doesn't play nicely with PDF of PNG. So you need to export as EPS, encapsulated postscript.

The answer is the same as for this question: Exporting BarChart3D as PDF produces artefacts - I'll repeat it here because you're asking about EPS export, and that question was about PDF export:

Map[SetOptions[#, 
    Prolog -> {{EdgeForm[], Texture[{{{0, 0, 0, 0}}}], 
       Polygon[#, VertexTextureCoordinates -> #] &[{{0, 0}, {1, 
          0}, {1, 1}}]}}] &, {Graphics3D, ContourPlot3D, 
   ListContourPlot3D, ListPlot3D, Plot3D, ListSurfacePlot3D, 
   ListVectorPlot3D, ParametricPlot3D, RegionPlot3D, RevolutionPlot3D,
    SphericalPlot3D, VectorPlot3D, BarChart3D}];

potential1 = 
 Plot3D[-3600. h^2 + 0.02974 h^4 - 5391.90 s^2 + 0.275 h^2 s^2 + 
   0.125 s^4, {h, -400, 400}, {s, -300, 300}, 
  PlotRange -> {-1.4*10^8, 2*10^7}, ClippingStyle -> None, 
  MeshFunctions -> {#3 &}, Mesh -> 10, 
  MeshStyle -> {AbsoluteThickness[1], Blue}, Lighting -> "Neutral", 
  MeshShading -> {{Opacity[.4], Blue}, {Opacity[.2], Blue}}, 
  Boxed -> False, Axes -> False]
Export["pot1.eps", potential1, ImageSize -> 500]

The first cell is there to add a Prolog to all your 3D plots that forces the conversion to bitmap upon export. Since this also applies to EPS, you'll get a bitmap wrapped in an EPS container that can be included in $\LaTeX$. The resulting EPS will look just like what you have posted.

Also note that you have to execute the first cell only one time, and all subsequent plots will look fine when exported to EPS or PDF.

Maybe you have an older version of $\LaTeX$ installed that doesn't play nicely with PDF of PNG. So you need to export as EPS, encapsulated postscript.

The answer is the same as for this question: Exporting BarChart3D as PDF produces artefacts - I'll repeat it here because you're asking about EPS export, and that question was about PDF export:

Map[SetOptions[#, 
    Prolog -> {{EdgeForm[], Texture[{{{0, 0, 0, 0}}}], 
       Polygon[#, VertexTextureCoordinates -> #] &[{{0, 0}, {1, 
          0}, {1, 1}}]}}] &, {Graphics3D, ContourPlot3D, 
   ListContourPlot3D, ListPlot3D, Plot3D, ListSurfacePlot3D, 
   ListVectorPlot3D, ParametricPlot3D, RegionPlot3D, RevolutionPlot3D,
    SphericalPlot3D, VectorPlot3D, BarChart3D}];

potential1 = 
 Plot3D[-3600. h^2 + 0.02974 h^4 - 5391.90 s^2 + 0.275 h^2 s^2 + 
   0.125 s^4, {h, -400, 400}, {s, -300, 300}, 
  PlotRange -> {-1.4*10^8, 2*10^7}, ClippingStyle -> None, 
  MeshFunctions -> {#3 &}, Mesh -> 10, 
  MeshStyle -> {AbsoluteThickness[1], Blue}, Lighting -> "Neutral", 
  MeshShading -> {{Opacity[.4], Blue}, {Opacity[.2], Blue}}, 
  Boxed -> False, Axes -> False]
Export["pot1.eps", potential1, ImageSize -> 500]

The first cell is there to add a Prolog to all your 3D plots that forces the conversion to bitmap upon export. Since this also applies to EPS, you'll get a bitmap wrapped in an EPS container that can be included in $\LaTeX$. The resulting EPS will look just like what you have posted.

Also note that you have to execute the first cell only one time, and all subsequent plots will look fine when exported to EPS or PDF.